Extensions of endomorphisms of $C(X)$

For a compact space $X$ we consider extending endomorphisms of the algebra $C(X)$ to be endomorphisms of Arens-Hoffman and Cole extensions of $C(X)$. Given a non-linear, monic polynomial $p\in C(X)[t]$, with $C(X)[t]/pC(X)[t]$ semi-simple, we show that if an endomorphism of $C(X)$ extends to the Arens-Hoffman extension with respect to $p$, then it also extends to the simple Cole extension with respect to $p$. We show that the converse to this is false. For a locally connected, metric $X$ we characterize the algebraically closed $C(X)$ in terms of the extendability of endomorphisms to Arens-Hoffman and to simple Cole extensions.