I'm looking for basic information about symmetry groups of discrete functions. It is difficult to search for such information, because searching for "symmetry group" gives results that refer almost exclusively to geometric figures or differential equations.
To be clear about what I'm asking about: given a function $f:X\to X$ for some finite (or countable) set $X$, an invertible function $g:X \to X$ is a symmetry of $f$ if $g \circ f\circ g^{-1} = f$. The symmetries of $f$ form a group under function composition.
I'm interested in any basic results about such symmetry groups. For example: given any arbitrary discrete group $G$, does there exist a discrete function $f$ such that $G$ is its symmetry group, or are the symmetry groups of functions a special subclass of groups?
I'm asking because I'm interested in discrete dynamical systems. I've only recently started (self-)learning group theory, but thinking about the symmetries of the state transition function seems a natural thing to do, so I'm looking for resources that can help me think in those terms.
I am particularly interested in the case where $f$ is a bijection.