Isologic groups with respect to fixed nilpotency class lower than theirs have equal nilpotency class

Statement
Suppose $$G_1$$ and $$G_2$$ are groups that are isologic groups with respect to the variety of groups of nilpotency class $$c$$. Then, the following are true:


 * 1) $$G_1$$ is a  nilpotent group if and only if $$G_2$$ is a nilpotent group
 * 2) Suppose $$G_1$$ is a group of nilpotency class at most $$d$$, with $$c \le d$$. Then, $$G_2$$ also has nilpotency class at most $$d$$
 * 3) Suppose $$G_1$$ is a group of nilpotency class exactly $$d$$, with $$c < d$$. Then, $$G_2$$ also has nilpotency class exactly $$d$$>

Related facts

 * Isoclinic groups have same nilpotency class