While studying the well-known "Geometry of Iterated Loop Spaces", I found this corollary which is not completely clear to me. (By $\mathcal{M}$ is meant the operad given by $\mathcal{M}(j):=\Sigma_j$, whith the other data suitably defined and such that its algebras are precisely the topological monoids).
More precisely, it is unclear to me why $\pi_2:\mathcal{C}\times \mathcal{M} \to \mathcal{M}$ should be $\Sigma$-equivariant. If so, I will certainly agree that any $E_{\infty}$ space is an $A_{\infty}$ space.
You have $(\mathcal{C} \times \mathcal{M})(j) = \mathcal{C}(j) \times \Sigma_j$, where $\Sigma_j$ acts on this space by the diagonal action (ie. $(c \times \tau) \sigma = (c\sigma) \times (\tau\sigma)$ for $c \in \mathcal{C}(j)$, $\tau \in \mathcal{M}(j)$ and $\sigma \in \Sigma_j$). Therefore $$\pi_2((c \times \tau) \sigma) = \pi_2((c \sigma) \times (\tau \sigma)) = \tau \sigma = \pi_2(c \times \tau) \sigma$$ and the morphism $\pi_2$ is equivariant.
Note that is is a general fact: any projection $\mathcal{A} \times \mathcal{B} \to \mathcal{A}$ (or $\mathcal{B}$) is a morphism of operads. I never used the specific form of $\mathcal{C}$ or $\mathcal{M}$ here.