Finitely many group extensions?

Question

Given a group $H$ and a finite group $F$. Consider the group extension

$1\to F\to G \to H \to 1$.

My question is, are there only finitely many isomorphism classes of $G$ satisfying the above exact sequence?

Answer 1

No. Take $F = \mathbb{Z}_2$ and

$$H = \bigoplus_{k \ge 1} \mathbb{Z}_{2^k}.$$

There are infinitely many non-isomorphic extensions of $H$ by $F$, namely the extensions

$$G_n = \bigoplus_{k \ge 1, k \le n - 1} \mathbb{Z}_{2^k} \oplus \mathbb{Z}_{2^{n+1}} \oplus \bigoplus_{k \ge n+1} \mathbb{Z}_{2^k}.$$

This follows from the result given in this math.SE question that it's possible to distinguish the number of summands isomorphic to $\mathbb{Z}_{2^k}$ in a direct sum of cyclic groups.

Answer 2

No. Let $p$ be a prime, let $F = C_p$, and let $H = \times_{k=1}^ \infty C_{p^k}$ be the direct product of cyclic groups of order $p^i$, with one factor for each $k > 0$.

Then there are infinitely many isomorphism classes of abelian groups $G$ satisfying the hypothesis, because $G$ could isomorphic to $$\times_{k=1}^{i-1}C_{p^k} \times C_{p^{i+1}} \times C_{p^{i+1}} \times_{k=i+2}^\infty C_{p^k}$$ for any $i>0$. (I'll leave it to the reader to prove that these possibilities are mutually non-isomorphic.)