Automorphisms of orthogonal decompositions and of group algebras of groups with partitions

This article is devoted to an investigation of the following conjecture. If $\{H_i\}$ is a family of subgroups that partition a finite group $G$, then every automorphism $\sigma$ of the group algebra $\mathbb C[G]$ that permutes the subalgebras $\mathbb C[H_i]$ also permutes the lines $\mathbb C\cdot g$, $g\in G$. The conjecture is confirmed for the following classes of groups with partitions: 1) Abelian groups; 2) non-Abelian 2-groups; 3) Frobenius groups with partitions inscribed in the standard partitions (consisting of the kernel together with all complements); 4) $HT$-groups; 5) $\operatorname{PGL}(2,q)$ and $\operatorname{PSL}(2,q)$; 6) the Suzuki groups $\operatorname{Sz}(2^{2k+1})$. This result confirms the conjecture concerning the finiteness of the automorphism groups of orthogonal decompositions constructed from the groups with partition occurring in the above list.