Finite solvable groups whose Gruenberg-Kegel graphs are isomorphic to the paw

The Gruenberg-Kegel graph (or the prime graph) 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 $G$. The paw is the graph on four vertices whose degrees are 1, 2, 2, and 3. We consider the problem of describing finite groups whose Gruenberg-Kegel graphs are isomorphic as abstract graphs to the paw. For example, the Gruenberg-Kegel graphs of the groups $A_{10}$ and $\mathrm{Aut}(J_2)$ are isomorphic as abstract graphs to the paw. In this paper, we describe finite solvable groups whose Gruenberg-Kegel graphs are isomorphic as abstract graphs to the paw.