Groups with an almost regular involution

An involution j of a group $G$ is said to be almost perfect in $G$ if any two involutions in $j^G$ whose product has infinite order are conjugated by a suitable involution in $j^G$. Let $G$ contain an almost perfect involution $j$ and $|C_G(j)|<\infty$. Then the following statements hold:
1) $[j,G]$ is contained in an $FC$-radical of $G$, and $|G:[j,G]|\leqslant|C_G(j)|$;
2) the commutant of an $FC$-radical of $G$ is finite;
3) $FC(G)$ contains a normal nilpotent class 2 subgroup of finite index in $G$.