= Solution
{c}
Assume first $n\ge2$. All <transpositions> form a single <conjugacy class> $T_1$. A <group automorphism> preserves <order of a group element>, so the image of a <transposition> is a nonidentity <involution>. By the order formula, its <disjoint cycle decomposition> contains only one-cycles and two-cycles, with at least one two-cycle. Thus it lies in some $T_k$, where $1\le k\le\lfloor n/2\rfloor$.
Moreover an <automorphism> maps the entire <conjugacy class> onto the <conjugacy class> of the image: for a fixed <transposition> $t$,
$$
\phi(gtg^{-1})=\phi(g)\phi(t)\phi(g)^{-1},
$$
and $\phi(g)$ runs over all of $S_n$ by surjectivity. Part (a)(i) identifies that latter <conjugacy class> as $T_k$. Hence $\boxed{\phi(T_1)=T_k}$, with equality rather than just inclusion. If $n=1$, $T_1$ is empty and the equality holds trivially with $k=1$.
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