Let $K$ be a local field with residue field $k$, let $E/K$ an elliptic curve of good reduction, $\tilde{E}/k$. The reduction map $E(K)\to \tilde{E}(k)$ respects addition, which leads to a natural injection $\operatorname{End}(E)\to\operatorname{End}(\tilde{E})$ whenever $\operatorname{End}(E)\cong \mathbb{Z}$.
But apparently, we always have an injection $\operatorname{End}(E)\to \operatorname{End}(\tilde{E})$, even when $E$ has complex multiplication. Lang, in his book on elliptic functions, says this follows from 'general reduction theory'.
Question: How does one go about constructing this injection, or even proving it exists?
I tried reducing the coefficients of the homogeneous polynomials, but that doesn't work, at least not for varieties in general. For example, consider the morphism $\mathbb{P}^1(\mathbb{Q}_2)\to \mathbb{P}^1(\mathbb{Q}_2)$ given by $(2X:Y)$. For reference, I'm using Silverman's definition of (projective) varieties, from his book on elliptic curves.
EDIT: This question is a near duplicate. However, I am interested in an explicit construction of the reductions of endomorphisms.