Suppose $A$ is a graded differential algebra, $h\subset g$ is an ideal, and that there is an $L_\infty$ action by $g/h$ on $A$. Is there any theorem that gives a quasi-isomorphism between the Lie-Hochschild cochain complex of $A$ with respect to the $L_\infty$ action by $g/h$, and a Lie-Hochschild cochain complex of the $L_\infty$ action of $g$ on $A$, that is invariant with respect to $h$, in some sense?
The particular example I'm interested in is when $A$ is the Hochschild cochain complex of $N=U(h)$, namely $C^p(N,N) = {\rm Hom}(\otimes^p N, N)$.
Sorry if the question is a little vague - part of what I'm looking for is the precise definition of this "Lie-Hochschild cohomology of $L_\infty$ action of $g$ relative to $h$".
