On finite groups generated by involutions

All groups in the abstract are finite. In theorem $1$ we prove that any group $A$, generated by $n$ involutions ($n \geq 3$), is a section $G/N$ of some group $B$, generated by three involutions (respectively, generated by an element of order $n$ and involution) in which $B/G$ is isomorphic $D_{2n}$ (respectively, $Z_n$). In theorem $2$ we consider the case when $A$ is a $2$-group. In theorem 3 and 4 we prove that any $2$-group is a section of a $2$-group generated by $3$ involutions and a section of a $2$-group generated by element of order $2^m$ and involution ($m$ may be arbitrary integer more than $1$). In the last part of the paper we construct some examples of $2$-groups, generated by $3$ involutions and of $2$-groups, generated by an element and involution of derived lengths $4$ and $3$ respectively.