Infinite groups of finite period

It is proved that there exist periodic groups containing an element of even order and only trivial normal $2$-subgroups in which every pair of involutions generates a $2$-group. This gives a negative answer to Question 11.11a in the Kourovka Notebook. Furthermore, we point out examples of finite simple groups that are recognizable by spectrum in the class of finite groups but not recognizable in the class of all groups.