Normal iff potential endomorphism kernel

Statement
The following are equivalent for a subgroup $$H$$ in a group $$G$$:


 * 1) $$H$$ is a normal subgroup of $$G$$.
 * 2) There exists a group $$K$$ containing $$G$$ such that $$H$$ is an endomorphism kernel in $$K$$.

Facts used

 * 1) Endomorphism kernel implies normal
 * 2) uses::Normality satisfies intermediate subgroup condition

(2) implies (1) (the easy direction)
If (2) holds, $$H$$ is a normal subgroup of $$K$$ by Fact (1). By Fact (2), then, $$H$$ is also a normal subgroup of $$G$$.

(1) implies (2) (the hard direction)
Let $$K$$ be the external direct product (we could also take the restricted external direct product) of $$G$$ and a countably infinite number of copies of $$G/H$$, i.e.,:

$$K = G \times G/H \times G/H \times \dots$$

Identify $$G$$ with the first direct factor (i.e., treating it as an internal direct product locally) and $$H$$ with the subgroup $$H$$ in the first direct factor.

By construction $$H$$ is the kernel of the endomorphism of $$K$$ that sends each direct factor to the next, with the first map $$G \to G/H$$ being the quotient map by $$H$$, and the remaining maps being identity maps. Explicitly, if $$\pi:G \to G/H$$ is the quotient map, the mapping we are talking about is:

$$(a,g_1,g_2,\dots,g_n,\dots)\mapsto (e,\pi(a),g_1,g_2,\dots,g_{n-1},\dots)$$

(where $$e$$ denotes the identity element).