Аннотация:

The Gruenberg-Kegel graph (or the prime graph) $\Gamma(G)$ of a finite group $G$ is the graph,
in which the vertex set is the set of all prime divisors of the order of $G$ and two different vertices $p$ and $q$ are adjacent if and only if there exists an element of order $pq$ in $G$. One of popular directions of research in finite group theory is the study of finite groups with given properties of their Gruenberg-Kegel graphs. In 2012-2013, the first author described finite groups with Gruenberg-Kegel graph as for the group $Aut(J_2)$ and as for the group $A_{10}$. The Gruenberg-Kegel graphs of groups $Aut(J_2)$ and $A_{10}$ are isomorphic (as abstract graphs) to the paw. The paw is the graph on four vertices whose degrees are 1, 2, 2, and 3. Generalizing the mentioned results of A.S. Kondrat'ev, we consider the problem of describing finite groups whose Gruenberg-Kegel graphs are isomorphic (as abstract graphs) to the paw. In four papers of 2018-2025, the authors considered the various cases of the problem. In this work, the authors continue the investigation of the problem and study its important new case when, for a finite non-solvable group $G$ whose Gruenberg-Kegel graph is isomorphic to the paw, the vertex of degree $3$ of the graph $\Gamma(G)$  is greather than 3.