Let $A=\oplus A_n$ be a primitively generated graded Hopf algebra, where each $A_n$ is a simplicial group. This allows us to define the homotopy group $\pi_*(A)$.
Question: is the graded Hopf algebra $\oplus \pi_*(A_n)\cong \pi_*(\oplus A_n)$ still primitively generated? Here * refers to a fixed positive integer.
Here is some context. Suppose one is given a filtration of a simplicial group. Then $A$ is often the associated graded algebra $E^0$ of this filtration. Usually then, for a spectral sequence, the $E^1$ will be $\pi_*(E^0)$.
Perhaps this question or similarly questions is generally known, and I would definitely be grateful for any references around this topic.
