Question: For a prime $p$, is every involution in $\mathbb{F}_p[[x]]$ a reduction modulo $p$ of some involution in $\mathbb{Z}[[x]]$?
Obviously, a reduction mod $p$ of an involution in $\mathbb{Z}[[x]]$ yields an involution in $\mathbb{F}_p[[x]]$, but do we get all involutions in $\mathbb{F}_p[[x]]$ this way?
Also, if you know of a reference where this is discussed, please let me know. Thank you.
Update: As noted by @quasi, $1+x$ is an involution modulo $2$, but it's not a reduction of any involution $f\in\mathbb{Z}[[x]]$, since $f(0)=0$ for any such $f$. So, a modified version of this question is: are all involutions in $\mathbb{F}_p[[x]]$ with a zero constant term reductions mod $p$ of involutions in $\mathbb{Z}[[x]]$?