  An involution over finite fields is a permutation polynomial whose inverse is
itself. Owing to this property, involutions over finite fields have been widely
used in applications such as cryptography and coding theory. As far as we know,
there are not many involutions, and there isn't a general way to construct
involutions over finite fields. This paper gives a necessary and sufficient
condition for the polynomials of the form $x^rh(x^s)\in \bF_q[x]$ to be
involutions over the finite field~$\bF_q$, where $r\geq 1$ and $s\,|\, (q-1)$.
By using this criterion we propose a general method to construct involutions of
the form $x^rh(x^s)$ over $\bF_q$ from given involutions over the corresponding
subgroup of $\bF_q^*$. Then, many classes of explicit involutions of the form
$x^rh(x^s)$ over $\bF_q$ are obtained.
