A representation is semisimple if its restriction to a subgroup of index prime to Char(F) is semisimple Let $G$ be a finite group and $H$ a subgroup whose index is prime to $p$. Suppose $V$ is a finite-dimensional representation of $G$ over $\mathbb{F}_p$ whose restriction to $H$ is semisimple. Prove that $V$ is semisimple.
 A: (Expanded form of the answers from the comments.)
A module is semisimple iff every submodule has a direct complement.
Aschbacher's Finite Group Theory on pages 39-40 proves that if U is a sub FG-module of V and U has an FP-module direct complement, then it has an FG-module direct complement where P is a Sylow p-subgroup of G.  The argument is an averaging argument as described.
Since H has index coprime to p, H contains a Sylow p-subgroup P of G.  Every semisimple FH-module is also semisimple as an FP-module (or just replace n in Aschbacher's proof with $[G:P]$ and allow $P=H$ to be any subgroup of index coprime to p), and so Aschbacher's result answers your question.

The same proof is phrased in more complicated language on pages 70,
71, and
72 of Benson's Representations and Cohomology Part 1.
A module is called relatively H-projective if G-module homomorphisms that split as H-module homomorphisms also split as G-module homomorphisms.  In other words, you just want to show that every G-module is relatively H-projective when $[G:H]$ is invertible in the ring, which is Corollary 3.6.9.
