Endomorphism join operator

Definition with symbols
Let $$p_1$$ and $$p_2$$ be two properties of endomorphisms of groups. Then, an endomorphism $$\alpha$$ of $$G$$ is said to satisfy the endomorphism join of $$p_1$$ and $$p_2$$ if:

There exist subgroups $$G_1$$ and $$G_2$$ of $$G$$, such that $$ = G$$ and such that $$\alpha$$ restricts to both $$G_1$$ and $$G_2$$ with the restriction to $$G_1$$ satisfying property $$p_1$$ and the restriction to $$G_2$$ satisfying property $$p_2$$.