Let $A$, $B$ be finite groups. Is it true that all short exact sequences $1 \rightarrow A \rightarrow A \times B \rightarrow B \rightarrow 1$ split on the right?


In other words, do there exist finite groups $A$, $B$ and homomorphisms $f: A \rightarrow A \times B$, $g: A \times B \rightarrow B$ such that $1 \rightarrow A \rightarrow A \times B \rightarrow B \rightarrow 1$ is exact and there does not exist a homomorphism $h: B \rightarrow A \times B$ such that $g \circ h = \text{id}\_B$?


An example when $A$, $B$ are not finite is given by $A = \prod\_{i=1}^\infty \mathbb{Z}$, $B = \prod\_{i=1}^\infty \mathbb{Z}/2\mathbb{Z}$, $f((n\_i)) = ((2n\_i),0)$, and $g((n\_i),(m\_i)) = (\overline{n\_1}, m\_1, \overline{n\_2}, m\_2, \ldots)$.



Do all exact $1 \to A \to A \times B \to B \to 1$ split for finite groups?
This is [true](https://doi.org/10.1515/JGT.2006.020) (1). It was extended to finitely generated profinite groups [here](https://doi.org/10.1515/JGT.2006.021) (2). Surprisingly, it is [also true](https://doi.org/10.1215/kjm/1250524308) in the category of finitely generated modules over a Noetherian commutative ring (3).


(1) *Joseph Ayoub, The direct extension theorem, J. Group Theory 9 (2006) 307-316*.


(2) *Goldstein, Daniel, Guralnick, Robert, The direct product theorem for profinite groups. J. Group Theory 9 (2006), no. 3, 317-322.*


(3) *Takehiko Miyata, Note on direct summands of modules. J. Math. Kyoto Univ.
Volume 7, Number 1 (1967), 65-69.* 


