2-groups with given properties of finite subgroups

A local finiteness is proved of 2-groups, all of whose finite subgroups (a) are nilpotent of class 2 or (b) belong to a variety defined by the law $[x,y]^2=1$. Besides, it is proved that the order of the derived subgroup of a 2-group $G$ is at most 2 if the order of every conjugacy class of every finite subgroup of $G$ is at most 2.