  Rota-Baxter operators on algebras, which appeared in 1960, have connections
with different versions of the Yang-Baxter equation, pre- and postalgebras,
double Poisson algebras, etc. In 2020, the notion of Rota-Baxter operator on a
group was defined by L. Guo, H. Lang, Yu. Sheng.
  In 2023, V. Bardakov and the second author showed that all Rota-Baxter
operators on simple sporadic groups are splitting, i. e. they are defined via
exact factorizations. In the current work, we clarify for which $n$, there
exist non-splitting Rota-Baxter operators on the alternating group
$\mathrm{A}_n$. For the corresponding $n$, we describe all non-splitting
Rota-Baxter operators on $\mathrm{A}_n$. Moreover, we describe Rota-Baxter
operators on dihedral groups $D_{2n}$ providing the general construction which
lies behind all non-splitting Rota-Baxter operators on $\mathrm{A}_n$ and
$D_{2n}$.
