I fix $C$ a symmetric monoidal model category (with a cofibrant unit if it helps). I'm assuming that it is closed, or at least that the tensor product commutes to colimits in each variable.
If $X$ is a monoid in $C$, and $A \to B$ is a map in $C$, I'm calling free extension of $X$ by $A \to B$ (along some map $A \to X$) the monoid object $X \coprod_{F(A)} F(B)$ where $F(A)$ and $F(B)$ are the free monoids on $A$ and $B$ and the pushout is a pushout of monoid object (using some arbitrary map $A \to X$).
It can be shown that if $A \to B$ is a (trivial) cofibration and $X$ is a cofibrant as an object of $C$, then the map $X \to X \coprod_{F(A)} F(B)$ is again a (trivial) cofibration in $C$.
The reason for this is that the map $X \to X \coprod_{F(A)} F(B)$ can be written as an $\omega$-composition of maps that are obtained by taking iterated "pushout-product" of $A \to B$ with itself and tensor product with $X$ (so in particular, are all cofibrations/trivial cofibrations as soon as $A \to B$ is).
What I wonder about is whether this can be generalized when "monoid" is replaced by $\mathcal{O}$-algebras for a general operad $\mathcal{O}$.
My guess after some computation is that something like the above can be done when $\mathcal{O}$ is $\Sigma$-cofibrant (i.e. cofibrant for the projective model structure on collections) and when $X$ is cofibrant for the projective model structure on $\mathcal{O}(1)$-object (assuming $X$ cofibrant in $C$ do not seem enough in general)
But I feel this might have been treated somewhere or that there might be other set of conditions under which something like this can be done.
