Given a finite ring $A$ (a finite group would be enough actually as we do not use multiplication of the ring). Is it possible to obtain the ring $R$ (with usual addition and multiplication being given by composition) of all maps (not structure presserving) $A \rightarrow A$ as a ring in GAP? Is it possible to get $R$ as an $K$-algebra in case $A$ is an $K$-algebra?
For example can do this in the special case of $A= Z_n$ the ring of all integers mod $n$.