On groups with an almost regular and almost perfect involution

In the article it is proved that a group with the least order of a Sylow 2-subgroup in the centralizer of almost perfect and almost regular involution $a$ is a soluble group (Theorem 2).
In addition, the study of the structure of the group $G$ with this almost perfect and almost regular involution $a$ with a Sylow 2-subgroup in $C_G(a)$ of least order among all these groups, which are not covered by Theorem 2, was initiated. It is proved that if $G$ is an essentially infinite group then this group $G$ is a soluble group (Theorem 3).
Let $G$ be an essentially infinite group. Let $a$ be an almost perfect involution in $G$. Let order of centralizer of this involution a be divided by 8, but the order of centralizer of this involution $a$ is not divided by 16. It is proved that if the center of the group $G$ does not have involutions then this group $G$ is a soluble group (Theorem 5).