Let $E$ be an elliptic curve of conductor $N\cdot p^2$ over $\mathbb{Q}$, defined by the equation
$$y^2=x^3+p^2b\cdot x + p^3\cdot c$$
and parametrized by a map
$$X_{0}(N\cdot {p}^{2})\rightarrow E$$
and let $E_{\sqrt(p)}$ be the curve
$$y^2=x^3+b\cdot x+c$$
parametrized by a map
$$X_{0}(N)\rightarrow E_{\sqrt(p)}$$
(Here, $p$ is a prime greater than or equal to $5$, and we assume that $E_{\sqrt(p)}$ reduces well at $p$.)

Both become isomorphic over $\mathbb{Q}[\sqrt{p}]$.
Is there a lift of that isomorphism to a morphism:
$$X_{0}(N\cdot {p}^{2})\rightarrow X_{0}(N)$$
or
$$X_{0}(N)\rightarrow X_{0}(N\cdot {p}^{2})$$
and if so, can that morphism be described explicitly?

If it is not possible in general, can we classify the cases where such a lift exists?