  We provide an answer, in a special but relevant case, to an open problem of
Drinfel'd by proving that the category of solutions of the set-theoretic
Yang-Baxter equation of Frobenius-Separability (FS) type is equivalent to the
category of pointed Kimura semigroups. As applications, all involutive,
idempotent, nondegenerate, surjective, finite order, unitary or indecomposable
solutions of FS type are classified. For instance, if $|X| = n$, then the
number of isomorphism classes of all such solutions on $X$ that are (a) left
non-degenerate, (b) bijective, (c) unitary or (d) indecomposable and
left-nondegenerate is: (a) the Davis number $d(n)$, (b) $\sum_{m|n} \, p(m)$,
where $p(m)$ is the Euler partition number, (c) $\tau(n) +
\sum_{d|n}\left\lfloor \frac d2\right\rfloor$, where $\tau(n)$ is the number of
divisors of $n$, or (d) the Harary number $\mathfrak{c} (n)$. The automorphism
groups of such solutions can also be recovered as automorphism groups
$\mathrm{Aut}(f)$ of sets $X$ equipped with a single endo-function $f:X\to X$.
We describe all groups of the form $\mathrm{Aut}(f)$ as iterations of direct
and (possibly infinite) wreath products of cyclic or full symmetric groups,
characterize the abelian ones as products of cyclic groups, and produce
examples of symmetry groups of FS solutions not of the form $\mathrm{Aut}(f)$.
