  Some families of linear permutation polynomials of $\mathbb{F}_{q^{ms}}$ with
coefficients in $\mathbb{F}_{q^{m}}$ are explicitly described (via conditions
on their coefficients) as isomorphic images of classical subgroups of the
general linear group of degree $m$ over the ring
$\frac{\mathbb{F}_{q}[x]}{\left\langle x^{s}-1 \right\rangle}$. In addition,
the sizes of some of these families are computed. Finally, several criteria to
construct linear permutation polynomials of $\mathbb{F}_{q^{2p}}$ (where $p$ is
a prime number) with prescribed coefficients in $\mathbb{F}_{q^{2}}$ are given.
Examples illustrating the main results are presented.
