  A subgroup $H$ of a group $G$ is said to be {\it pronormal} in $G$ if $H$ and
$H^g$ are conjugate in $\langle H, H^g \rangle$ for each $g \in G$. Some
problems in Finite Group Theory, Combinatorics, and Permutation Group Theory
were solved in terms of pronormality, therefore, the question of pronormality
of a given subgroup in a given group is of interest. Subgroups of odd index in
finite groups satisfy a native necessary condition of pronormality. In this
paper we continue investigations on pronormality of subgroups of odd index and
consider the pronormality question for subgroups of odd index in some direct
products of finite groups.
  In particular, in this paper we prove that the subgroups of odd index are
pronormal in the direct product $G$ of finite simple symplectic groups over
fields of odd characteristics if and only if the subgroups of odd index are
pronormal in each direct factor of $G$. Moreover, deciding the pronormality of
a given subgroup of odd index in the direct product of simple symplectic groups
over fields of odd characteristics is reducible to deciding the pronormality of
some subgroup $H$ of odd index in a subgroup of $\prod_{i=1}^t \mathbb{Z}_3\wr
Sym_{n_i}$, where each $Sym_{n_i}$ acts naturally on $\{1,\dots, n_i\}$, such
that $H$ projects onto $\prod_{i=1}^t Sym_{n_i}$. Thus, in this paper we obtain
a criterion of pronormality of a subgroup $H$ of odd index in a subgroup of
$\prod_{i=1}^t \mathbb{Z}_{p_i}\wr Sym_{n_i}$, where each $p_i$ is a prime and
each $Sym_{n_i}$ acts naturally on $\{1,\dots, n_i\}$, such that $H$ projects
onto $\prod_{i=1}^t Sym_{n_i}$.
