Background:
Given $\mathbb{F}_2$-graded Hopf algebra $A$, define as $h_A(d) := \dim A_d$ (where the dimension is as a $\mathbb{F}_2$-vector space), where $A_d$ is the degree $d$ component of $A$. The Poincare series $P_A(t)$ is then $\sum_{t = 0}^\infty h_A(d)t^d$.
Then, given $P_A(t)$ and $P_B(t)$, how do these relate to $P_{A//B}(t)$, where $A//B = A \otimes_B \mathbb{F}_2$?
It seems that $P_{A/B}(t)$ would be $P_A(t) - P_B(t)$ or $P_A(t)/P_B(t)$. However, neither seem to be true. Indeed, quotienting by $B$ invokes lots of relations which don't seem to be adequately cancelled by just subtracting.