It may be possible in rare circumstances, but the natural thing is that mathcalO(H) is a quotient of mathcalO(G), not a subalgebra. The quotient map is dual to the inclusion HtoG.
I guess in the case that G is a direct product HtimesK, you get
mathcalO(G)simeqmathcalO(H)otimesmathcalO(K),
and in this case you have what you want. But in general, the arrow should go the other direction.
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