  In this work we study orthogonal polynomials via polynomial mappings in the
framework of the $H_q-$semiclassical class. We consider two monic orthogonal
polynomial sequences $\{p_n (x)\}_{n\geq0}$ and $\{q_n(x)\}_{n\geq0}$ such that
$$ p_{kn}(x)=q_n(x^k)\;,\quad n=0,1,2,\ldots\;, $$ being $k$ a fixed integer
number such that $k\geq2$, and we prove that if one of the sequences $\{p_n
(x)\}_{n\geq0}$ or $\{q_n(x)\}_{n\geq0}$ is $H_q-$semiclassical, then so is the
other one. In particular, we show that if $\{p_n(x)\}_{n\geq0}$ is
$H_q-$semiclassical of class $s\leq k-1$, then $\{q_n (x)\}_{n\geq0}$ is
$H_{q^k}-$classical. This fact allows us to recover and extend recent results
in the framework of cubic transformations, whenever we consider the above
equality with $k=3$. The idea of blocks of recurrence relations introduced by
Charris and Ismail plays a key role in our study.
