i assume that the question you are asking can be formalized in the following way: given two finite groups G and H, how many pairs (A, B) are there with A subgroup G and B subgroup H and A is isomorphic to B?

the first clear limitation is that the subgroup orders |A| = |B| must divide gcd(|G|, |H|), but that is a very weak condition (though it can be enough, e.g., when the gcd is 1)

a simpler first cut might be the case G = H, which is to find the number of isomorphic pairs of subgroups of a given finite group, but i don't think this is even known for all p-groups (groups G with |G| = power of a prime p)

another special case is for abelian (commutative) groups, for which the problem can be reduced to the various p-groups, and, i suspect, a complete answer can be written

I don't think that this is a typical question, one can refine the question considerably (for instance take G =H to be an arbitrary p-group, and try find the number of subgroups of order p( which are all isomorphic)), and one is still far from a satisfactory answer.

Let G and H be two groups.Let f be a map from G to H.Let S be all subgroups of G

containing kernel f. Let M be the set of all sub groups of H. Let g be a map from S to T. Then it can be shown that g is one -one and on to.

If  G IS A CYCLIC GROUP   ( 1) IF ORDER   OF G IS FINITE SAY n   THEN  ODER  OF  AUTO(G)=ϕ(n)

                                             (2) IF ORDER G IS INFINITE THEN  THE ORDER OF 

    AUTO(  G)=    2