I just read about permutation groups. Before going further this question came up in my mind.
Isn't every group a permutation group?
The definition says, "one-to-one mappings of a set onto itself is called a permutation". We say the group has a binary operation which also is a one-to-one mapping onto itself.
So, the question is, why do we need another name to the same thing?
On
the term "permutation group" is really a misnomer as you suggest. on the other hand "group of permutations" is more acceptable in the same way as "group of rotations of the 3-sphere" or "group of rational functions" define a group from a particular presentation. however the group itself is an abstract structure - an isomorphism class
As the comments point out, it is true that every group $G$ acts on some set as a permutation: one such action is left multiplication of $G$ on itself; and perhaps $G$ has many different such actions.
But the power of the "group" concept is that it is an abstraction. This allows us to recognize many more groups when we first encounter them, even if their first appearance is not as a permutation group.
Important examples of this phenomenon are the various groups that come up in topology, such as fundamental group $\pi_1(X,p)$ of a path connected topological space $X$ with base point $p$. The elements of $\pi_1(X,p)$ are path homotopy classes of closed paths based at $p$. The group operation is induced by concatenation of paths. This is not a description that seems to have anything to do with permutations.
Eventually we do learn that to study $\pi_1(X,p)$ we must study its deck transformation action on the universal covering space of $X$, which is an elaborate permutation action. However that is not how we first encounter fundamental groups. Our study of fundamental groups would be inhibited if we refused to recognize their existence based on the fact that they are not given to us as permutation groups.