TITLE: Constituents of induced representation
QUESTION [9 upvotes]: Let $H < G$ be finite groups with $|G:H|=n$, and let $M$ be an irreducible $FH$-module for some field $F$. Is it always true that all irreducible constituents of the induced $FG$-module $M^G$ have dimension at least $\dim M$? If not, then is it true that the composition length of $M^G$ is at most $n$?
For complex representations, these statements follow from Frobenius Reciprocity. But do they remain true for modular representations? I am particularly interested in the case when $F$ is a finite prime field.

REPLY [2 votes]: In the case that $n$ is prime and $H$ is normal in $G$. Your second questions has an affirmative answer. The composition length is $1$, $n$, or $1+d$ where $d\mid (n-1)$, in each case at most $n$. See the main theorem from:
S.P. Glasby and L.G. Kovács,
Irreducible modules and normal subgroups of prime index.
Comm. Algebra 24 (1996), no. 4, 1529–1546.