  Given a prime power $q$ and positive integers $m,t,e$ with $e > mt/2$, we
determine the number of all monic irreducible polynomials $f(x)$ of degree $m$
with coefficients in $\mathbb{F}_q$ such that $f(x^t)$ contains an irreducible
factor of degree $e$. Polynomials with these properties are important for
justifying randomised algorithms for computing with matrix groups.
