On a problem of Gaschütz

Recall that a formation is a class of finite groups (up to isomorphism) that is closed under taking quotient groups and (pairwise) subdirect products. For a group G let form(G) denote the smallest formation containing G. There was a conjecture (the Gaschütz problem) that form(G) contains only finitely many subformations, for any G. This was proved in the case where G is solvable, and in some other cases. In the article a counterexample is constructed. Namely, if A=2S5 (a double cover of S5), then form(A) has infinitely many subformations. The precise structure of the lattice of subformations of form(A) is found. The main technical tool is a reduction to the problem of finding submodule structure for some module over a certain category.