Suppose $Q$ is a filtered algebra, with associated graded algebra $\text{gr}(Q)$. If we have an action of a ring $R$ on $\text{gr}(Q)$ (i.e. $\text{gr}(Q)$ is an $R$-module) then it seems clear that, under a choice of isomorphism between $\text{gr}(Q)$ and $Q$, $Q$ will also be an $R$-module. (I am interested in situations when $Q$ is a PBW-type deformation of $\text{gr}(Q)$ so a natural choice of such an isomorphism exists).
However what if $\text{gr}(Q)$ is an $H$-module algebra, for a Hopf algebra $H$. So in addition to being an $H$-module (as above), we also have that the product of $\text{gr}(Q)$ is an $H$-module homomorphism. In particular we have: $h\rhd 1=\epsilon(h)1$ and $h\rhd (a\cdot b)=(h_1\rhd a)\cdot (h_2\rhd b)$. When will the $H$-module structure on $Q$ (coming from $\text{gr}(Q)$) also make $Q$ an $H$-module algebra?
I can't find anything in the literature about this so any references/advice would be great.