= Solution
Let $g:A\to B$ be a finite partial elementary map in $M$. In a countable language, the algebraic closures of finite sets are countable. An alternating back-and-forth construction extends $g$ to an isomorphism
$$
\operatorname{acl}(A)\cap M
\longrightarrow
\operatorname{acl}(B)\cap M.
$$
At each step, an element algebraic over the current domain has a finite algebraic type, and elementarity provides a matching realization on the other side.
Choose a <basis of a pregeometry> $C$ over the first closed set and a <basis of a pregeometry> $D$ over the second. The <dimension of a pregeometry> is the same for the two extensions: the isomorphism preserves the finite ranks already contributed by $A$ and $B$, while both sides have the same ambient dimension. Choose a bijection $C\to D$. Part (a) and uniqueness of the <generic type in a strongly minimal theory> make the enlarged map elementary. Since an elementary submodel is algebraically closed in the monster,
$$
M=\operatorname{acl}(A\cup C)=\operatorname{acl}(B\cup D),
$$
and the map extends over algebraic closure to an automorphism of $M$. Thus every finite partial elementary map extends to an automorphism: every model is homogeneous, as stated in <homogeneity of models of a countable strongly minimal theory>.
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