Let $A$ be a cocommutative bialgebra object (or even a Hopf algebra) in a symmetric monoidal category. Define an operad $\mathtt{P}_A$ by $\mathtt{P}_A(r) = A^{\otimes r}$ (so that $\mathtt{P}_A(0) = 1$ is the unit of $\otimes$), and the composition is given by the following formula: $$\gamma(a_1 \otimes \dots \otimes a_r; \underline{b}_1, \dots, \underline{b}_r) := (a_1 \cdot \underline{b}_1 \otimes \dots \otimes a_r \cdot \underline{b}_r)$$ where $a_i \in A$, $\underline{b}_j \in A^{\otimes k_j}$, and $A^{\otimes k}$ becomes an $A$-module using the cocommutative diagonal of $A$, and using the counit for $A^{\otimes 0} = 1$ (I'm writing this formula in the case of vector spaces, I don't want to draw a humongous commutative diagram; I hope the general definition is clear).
Is this construction an example of a more general phenomenon / can it be described differently? It appears when defining semi-direct products of operads, for example. It feels like something rather simple that appears naturally (I don't know, maybe the free construction on something), but I can't write it as an application of a more general construction; I realize it's the semi-direct product $\mathtt{Com} \rtimes A$, but it feels a bit circular.