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Children who have reached the age of Chinuch are to be educated in the Mitzvah of Sefiras HaOmer. They are to recite it at night with a blessing [if they have not missed a previous night], following the same laws as adult men. The age of Chinuch is from the age that the child understands the meaning of Sefiras Haomer.
In areas that night begins very late, must one keep the child awake until late at night in order to say Sefiras Haomer?
It is proper to wake the child up after night in order to say Sefira. This applies only if the child has reached the age of Chinuch. If this poses difficulty, one may simply recite Sefira with the child during the day.
 Admur 343/2-3 regarding all Mitzvos, and in accordance to 1st opinion in Admur 70/2; Shaareiy Teshuvah 489/20 “Our custom is that children who have reached the age of Chinuch count with a blessing”; Maharil Hilchos Sefiras Haomer; Regarding Mitzvos that apply in early morning or at night, during hours a child might not be awake-see Admur 70/2 regarding Shema that this matter is disputed in Poskim although concludes and that one should follow the first, stringent, opinion; M”B 426/1 that states this dispute regarding Shema would likewise apply regarding Kiddush Levana, and accordingly, the same should apply regarding Sefiras Haomer. To note, that according to those who rule that the Mitzvah of Sefira applies also by day, this dispute is irrelevant, and one is obligated to educate his child in Sefira according to all.
 As is the law by an adult; However see Nitei Gavriel 24/7 footnote 13 who novelizes that they may continue to count with a blessing.
 Admur 343/3
 See Admur 70/2 regarding Shema