  The aim of this article is to define and study a notion of unstable algebra
over an operad that generalises the classical notion of unstable algebra over
the Steenrod algebra. For this study we focus on the case of characteristic 2.
We define $\star$-unstable $\mathcal P$-algebras, where $\mathcal P$ is an
operad and $\star$ is a commutative binary operation in $\mathcal P$. We then
build a functor that takes an unstable module $M$ to the free $\star$-unstable
$\mathcal P$-algebra generated by $M$. Under some hypotheses on $\star$ and on
$M$, we identify this unstable algebra as a free $\mathcal P$-algebra. Finally,
we give some examples of this result, and we show how to use our main theorem
to obtain a new construction of the unstable modules studied by Carlsson,
Brown-Gitler, and Campbell-Selick, that takes into account their internal
product.
