This is not exactly an incarnation of the question you asked, in the sense that is not so much an action of a Galois group but rather an action whose existence is governed by a Galois group of number-theoretic origin, but it seems likely to be of interest.

Let $K$ be a number field, and let $K^{(1)}$ be the maximal unramified abelian extension of $K$.  The Galois group of $K^{(1)}/K$ is a subquotient of  Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$) which is isomorphic to the class group of $K$.  Note that by Minhyong Kim's answer <a href="https://mathoverflow.net/questions/49960/are-class-numbers-encoded-in-the-absolute-galois-group-of-mathbb-q/49967#49967">here</a>, we can characterize this subquotient purely Galois-theoretically.  Several authors have discovered surprising links between the arithmetic of number fields and actions of groups on spheres.  In particular, when $K$ is the real cyclotomic field $K_m=\mathbb{Q}(\zeta_m+\zeta_m^{-1})$, the class group appears to govern the free actions of binary dihedral groups on spheres $S^n$ with $n\equiv 3\pmod{4}$.  Let me loosely quote/paraphrase from Lang's "Units and Class Groups in Number Theory and Algebraic Geometry" (bolding mine):

<blockquote>
C. T. C. Wall has already shown to depend in part on the
2-primary component of the ideal class group in real cyclotomic fields $K_m^+$ for suitable $m$...Using the algebraic background of a paper
of Wall, applied to the surgery exact sequence, Thomas gives examples
for the binary dihedral group $D_{4p}$ of order $4p$ operating freely on $S^{4k-1}$ with $k\geq 2$, <b>when the order of</b> $[(K_p^+)^{( 1)}:K_p^+]$ <b>is odd.</b>
<br>
<p>...
<br>
Furthermore, according to Thomas, there exist free actions by $D_{4p}$ which
can be topologically distinguished <b>only</b> by an invariant in the 2-primary part
of the ideal class group of $K_p^+$.
</blockquote>

Perhaps needless to say, the study of these degrees $[(K_p^+)^{( 1)}:K_p^+]$, even their 2-part, is of tremendous interest in algebraic number theory (Vandiver's conjecture, etc.), so the link to actions on spheres is surprising.