A child who is transgressing for the sake of another Jew:[1]
If a child [of any age[2]] is committing a transgression for the benefit of any other Jew[3] who is already Bar Mitzvah[4]—for example, extinguishing a fire in that Jew’s home when there is no danger to human life—then every Jew[5] is [Rabbinically[6]] obligated to stop the child or reprimand him. This responsibility is not restricted solely to the child’s father or legal guardian. [This obligation to protest a child’s actions applies even if the child performed the act privately and inconspicuously, with no one else present. In such a case once it becomes known that the act was performed on behalf of an adult, one is then obligated to protest and object to the child’s conduct.[7]]
A Rabbinical prohibition: See Halacha 8D regarding if one must protest a child who is performing a Rabbinical prohibition on behalf of an adult.
[1] Admur 343:1 “If he does so on behalf of an adult, one must protest and stop him, by rabbinic law, as explained in סימן של״ד.”; Admur 334:26 in parentheses “A child, however, who comes to extinguish a fire must be stopped, even if he has not yet reached the age of chinuch and even if his father is not present. This is because a child does not possess the independent judgment necessary to act for his own benefit. Rather, since he knows that the fire is causing a loss to his father (or to another Jew), we assume that he is extinguishing it on the Jew’s behalf in order to please him.”; Admur 362:17 “One should not create a human partition (mechitzah shel bnei adam) except in a time of need and pressing circumstances. If one forgot an item in a public domain, it is preferable to bring young children there and allow them to pick up the item on their own and bring it into the house than to construct a human partition and have an adult bring it in. This applies only when the child does not know that the object belongs to his father. In that case, the child is acting for his own benefit, thinking that he has found an object in the public domain. However, if the child knows that the object belongs to his father, then he is presumably bringing it for his father’s benefit, and every person is obligated to stop the child from bringing it, as explained in Siman 334. Certainly, one may not bring the child there in order that he retrieve the object. The same law applies if the object belongs to another person and the child recognizes it and intends to bring it to that individual. All of the above applies to other people. However, a father is obligated to educate his child, rebuke him, and prevent him from desecrating Shabbos whenever he sees him doing so, even when the child is acting for his own benefit and even regarding rabbinic prohibitions, as explained in Siman 343. Certainly, a father may not bring the child to the location in order that he retrieve the object, even if the object does not belong to the father.”; Michaber 334:25; Beis Yosef 334; Rosh 1:1; Rabbeinu Yerucham 1:13; Chayeh Adam 66:4; Kaf Hachaim 343:1; Piskeiy Teshvuos 343:5, 13; 343:2 footnote 22; Toras Hakatan Chapter 31; Chinuch Habanim Kehilchaso Chapter 45:20-40
[2] Admur 334:26 “Even if he has not reached the age of Chinuch”; Bach 334 “This child is prior to the age of Chinuch , such as a four or five year old and we are not certain as to whether this Katan is performing the action on behalf of his father”; Tosafus Shabbos 121a
A child below the age of comprehension: This obligation applies only to a child who has reached a level of mental maturity such that he is capable of acting on behalf of others. In such a case, when it is determined that the child performed the act for the benefit of an adult, there is an obligation to protest his actions even if he is officially below the age of Chinuch. However, if the child is so young that he lacks the comprehension to act on behalf of others, then even if it appears outwardly that he is doing so, there is no obligation to object. Thus, for example, if a two‑year‑old plays with a light and turns it on or off—even if the adults had been discussing whether they wanted the light on or off—there is no requirement to protest. Since a child of that age does not possess sufficient understanding to perform actions on behalf of others, his conduct is not treated as such. [See Michaber 334:25; Rishonim on Shabbos 121a; Chikrei lev 63; Piskeiy Teshuvos 343:5 footnote 74]
[3] Admur 343:1 “Bishvil Gadol”; 334:26 “on behalf of his father (or another Jew)”; 362:17”; M”B 334:65 in name of P”M; Ketzos Hashulchan 147 footnote 11
Father versus other person: So rules also Admur in 334:26 that if the child is going to extinguish the fire even on behalf of a person other than his father, he must be reprimanded. Vetzaruch Iyun why Admur in 334:26 stated this addition (another person) in parentheses. To note however that in the Gemara [Yevamos 104a] explains that the reason the child must be reprimanded is because he is doing so on behalf of his father, and thus perhaps here Admur novelizes that even when done on behalf of others he must be reprimanded. [Ketzos Hashulchan 147 footnote 11] This is also explicitly implied in Rashi there which equates this case to the case brought earlier in the Gemara regarding having children carry keys in a public domain on behalf of another person, of which there it was allowed because they were not doing so on behalf of that person. Nevertheless, tzaruch Iyun why the need to place this addition in parentheses. The Mishneh Berurah [334:65] rules in the name of the Peri Megadim as above that the child is to be reprimanded even when he goes to extinguish another person’s fire.
Other Opinions: See Chinuch Habanim Kehilchaso Chapter 45:20-40
[4] See Chanoch Lanaar 7:3 in name of Sefer Malei Haroim that if a child is doing so on behalf of another child he does not need to be protested. So is also implied from Admur ibid which states “Bishvil Gadol”; Piskeiy Teshuvos 343:2 footnote 22
[5] Implication of Admur 343:1-2 and 334:26 “even if his father is not there” and in parentheses “or for another Jew”; See Admur 343:2 that this obligation to reprimand and correct improper behavior is not limited to the Beit Din, even though they are the primary subject discussed in this halacha. Rather, this obligation applies to every individual. This is made explicit by the wording in the beginning of the next halacha, where Admur states that all of the above applies when dealing with “other people”, clearly indicating that the responsibility is not confined to judicial authorities alone.
[6] A father regarding Shabbos: Some Poskim rule that regarding Shabbos a father is Biblically obligated to prevent his son from transgressing on his behalf, as the verse states Lo Saseh Kol Melacha, Ata Ubincha Ubitecha. [See Shaar Hatziyon 343:54; Biur Halacha 266:6; Ketzos Hashulchan 141 footnote 28; Mishkanos Yaakov O.C. 118; Piskeiy Teshuvos 343:3 and 5 footnotes 32, 61 and 67-70] Practically, Admur and other Poskim do not agree with this opinion. [See Admur 266:10; and so is Setimas Admur throughout 343; Poskim in Piskeiy Teshuvos ibid footnote 70]
[7] See Admur 334:26 “Even if his father is not there”; Beis Yosef 334 based on Rashi Shabbos 121a; M”B 334:65; Piskeiy Teshuvos 343:5 footnote 72