Let $G$ be a discrete countable group which acts on a countable set $X$. This action defines a standard unitary representation, called the permutation representation, on the Hilbert space $l_2(X)$ of all functions $X\to \mathbb C$ with finite $l_2$ norm. The Hilbert space have an obvious Hilbert basis consisting of elements $\delta_x$ indexed by $x\in X$ ($\delta_x$ is the function which is 1 on $x$ and 0 everywhere else), and $G$ acts on the basis by permutations.
I have a fairly general question.
When is it possible to deform the permutation representation to some non-conjugate representation of $G$ in $l_2(X)$?
Here "non-conjugate" means that the deformation is not trivially realized by an action of the isometry group of $l_2(X)$. More generally, I would like to know if there is a theory of infinitesimal deformations which may produce non-conjugate representations starting from a permutation representation for specific pairs $(G, X)$; I suppose that, since $G$ is not compact, a general answer is unknown, but it might be known for specific cases. Any reference is welcome.