By a splitting theorem I mean a statement of the type: if $G$ has a normal subgroup $N$ and some hypotheses are satisfied, then there is a complement, that is a subgroup $Q$ of $G$ such that $G = N \rtimes Q$.
Examples of this are the Schur-Zassenhaus Theorem, where the condition is that $|N|$ and $|G/N|$ be coprime; or the Gaschütz theorem, which reduces the splitting over an abelian $p$-group to the splitting of a $p$-Sylow subgroup of $G$ over the same $p$-group.
In general, there always seem to be statements about prime divisors, and I cannot find any statement in the $p$-group case, where none of the above applies. Is there any general statement about splittings in finite $p$-groups? Or is there some theory that has been developed to deal with what looks like the hardest case?
I would be happy even with a statement about abelian $p$-groups.