Say $\ell^2-\ell-\delta\equiv0\bmod p$ is a polynomial $x^2-x-\delta$ with root $\ell$ and we lift to $\ell'^2-\ell'-\delta\equiv0\bmod p^2$ by Hensel lifting my question is following: if $g^y\equiv \ell^2\bmod p$ held at a generator $g$ then is there an explicit generator $g'$, which can be found in polynomial time, for which $$g'^y\equiv\ell'^2\bmod p^2$$ holds true?



A problem on generators and Hensel lifting
Write $g'=g(1+ap)$ so that
$$g'^y\equiv g^y(1 + pya)\pmod{p^2}.$$
It follows that we can take
$$a = \frac{(\ell^2/g^{y}\bmod p^2)-1}{py},$$
which can be computed in polynomial time.


