As a consequence of THEOREM 2.6 in [ROBERT J. BLATTNER, SUSAN MONTGOMERY, CROSSED PRODUCTS AND GALOIS EXTENSIONS OF HOPF ALGEBRAS, PACIFIC JOURNAL OF MATHEMATICS Vol. 137, No. 1, 1989, 37-54], we know that:

Let $p$ be a prime, $H$ be a normal subgroup of $G$ with $(p,|G:H|)=1$ and $V$ be a left $kG$-module, where the characteristic of $k$ is $p$. If $V$ is a semisimple $kH$-module, then $V$ is a semisimple $kG$-module.

This consequence tells us that sometimes the complete reducibility can be deduced from the restricted representation.

Question: Besides the above consequence, are there any other similar theorems?