Let $X$ be a smooth projective scheme over a field $\mathbf{k}$ of characteristic zero such that $\mathrm{H}^0(X, \mathrm{T}X)$ vanishes, and let $f$ be an automorphism of $X$. I would like to have an explicit example of such a pair $(X, f)$ such that:

1) The automorphism $f$ lifts to some non trivial first order deformation $\mathfrak{X}$ of the scheme $X$, that is some nonzero element of $\mathrm{H}^1(X, \mathrm{T}X)$ is fixed by the action of $f$.

2) Given any nontrivial deformation $\widetilde{\mathfrak{X}}$ of $X$ over a small extension $A$ of $\mathbf{k}[t]/t^2$ such that the pullback of $\widetilde{\mathfrak{X}}$ is $\mathfrak{X}$, then $f$ doesn't lift to $\widetilde{\mathfrak{X}}$.

edit: by nontrivial, I mean that $\widetilde{\mathfrak{X}}$ is not obtained by pull back from $\mathfrak{X}$ via a section of the map $A \rightarrow \mathbf{k}[t]/t^2$.