Does a group action always induce a quotient?

Question

Let $G$ be a group, let $X$ be a set on which $G$ acts (possibly non-faithfully).

I would be tempted to say the following:

There exists a normal subgroup $K$ of $G$, such that:

• $G/K$ acts faithfully (or freely? or both?) on $X$, and
• The action of $G$ and of $G/K$ on $X$ are isomorphic.

Is this true? If yes, what is its relation to stabilizers? If no, are there conditions to require in order to make it true? Or is it simply false in general (for example, $K$ would depend always on the point of $X$)?

A reference would also be welcome, but please, for general sets (not modules).

Thanks!

Answer 1

Hint

The action can be written as a homomorphism $\phi : G \to S(X)$, where $S(X)$ is the symmetric group over $X$. Now take $K$ as the kernel of $\phi$.