  Let $\mathbb{F}_q[T]$ be the polynomial ring over a finite field
$\mathbb{F}_q$. We study the endomorphism rings of Drinfeld
$\mathbb{F}_q[T]$-modules of arbitrary rank over finite fields. We compare the
endomorphism rings to their subrings generated by the Frobenius endomorphism
and deduce from this a refinement of a reciprocity law for division fields of
Drinfeld modules proved in our earlier paper. We then use these results to give
an efficient algorithm for computing the endomorphism rings and discuss some
interesting examples produced by our algorithm.
