This is directly from Dummit & Foote (pg. 118)
We now consider a generalization of the action of a group by left multiplication on the set of its elements. Let $H$ be any subgroup of $g$ and let $A$ be the set of all left cosets of $H$ in $g$. Define an action of $G$ on $A$ by
$ g \cdot aH=gaH$ for all $g \in G, aH \in A$ where $gaH$ is the left coset with representative $ga$.
Is the action permuting the left cosets of $H$ in $G$?
For what amounts as probably an overkill answer, realize the following:
You can define the notion of a group action on any category $\mathcal C$ as follows:
If $X \in \mathrm{Obj}(\mathcal C)$, define $\mathrm{Aut}(X)$ to be the collection of endomorphisms which are also isomorphisms. The group operation is composition, the endomorphism property ensures we can compose any two elements, the definition of a category ensures the existence of an identity and associativity of composition, and the isomorphism property ensures the existence of an inverse.
A group action of $G$ on an object $X \in \mathrm{Obj}(\mathcal C)$ is then just a group homomorphism $G \to \mathrm{Aut}(X)$. The way that we normally think about this is as $\psi: G \to \mathrm{Aut}(X)$ defines a group action by $g\cdot x = \psi(g)(x)$.
Now for any set $X$, you can define the permutation group $S_X$. The set-automorphisms can then be shown to be a subgroup of $S_X$; that is, $\mathrm{Aut}_{\textbf{Set}}(X) \subseteq S_X$, so if you prefer you may think of $\psi: G \to \mathrm{Aut}(X)$ as $\psi: G \to S_X$.
Thus the answer is affirmative, the group action permutes the cosets. More generally, every group action on a concrete category acts by permtuations.