A group action up to a function.

Question

Let $G$ be a group and an "action" $\psi$ of $G$ on a set $\Delta$ inside a vector space $M$, of the form: $$\psi(g)\psi(h)v=(\psi(gh)+\phi(g,h))v$$ for $g,h\in G$, $v\in \Delta$ and $\phi$ a function on $G\times G$. There is a name for this kind of group action up to a function? clearly is not a group action, traditionally speaking, however it came up when studying actions on subsets from a well-defined group action. So, if $\Delta\subset M$ and $G$ acts on $M$, the projection to an arbitrary subset has been studied?