Let $Q\in{\mathbb Z}[X], Q=\sum_{k=0}^m q_kX^k$, with degree $m$ and Galois group $S_m$ (over $\mathbb Q$). Consider the evenized-reciprocalized polynomial $P(X)=X^{2m}Q\bigg(X^2+\frac{1}{X^2}\bigg)$.
Is it true that, except on a set of Zariski dimension $0$ (I mean, in terms of the coefficients $q_0,q_1,\ldots, q_m$), the Galois group of $P$ (over $\mathbb Q$) is $V_4 \wr S_m$ ?
This question is natural in the context of another recent question of mine.