= Solution
Choose an integral <Weierstrass equation of an elliptic curve> with unit <elliptic-curve discriminant>. For $P\in E(\mathbb Q_p)$, choose homogeneous coordinates $[X:Y:Z]$ in $\mathbb Z_p$ with at least one unit by multiplying all three coordinates by a common power of $p$. Define
$$
\operatorname{red}(P)=[\overline X:\overline Y:\overline Z]\in\widetilde E_p(\mathbb F_p).
$$
Two primitive integral representatives differ by a unit, so this is well-defined. Every point of the reduced cubic is a <nonsingular point of an algebraic curve>, and the identity reduces to $[0:1:0]$. This definition includes points whose affine coordinates are nonintegral.
To prove the <good-reduction map respects elliptic addition>, use the smooth proper integral model. The <elliptic curve group law> is a morphism on this model, constructed by addition of degree-zero <divisor classes>; it is defined even where an affine chord formula has a vanishing denominator. Passing to the <residue field> commutes with this morphism. Hence
$$
\operatorname{red}(P+Q)=\operatorname{red}(P)+\operatorname{red}(Q),
$$
so reduction is a <homomorphism of abelian groups>.
For surjectivity, the simple-root <Hensel lemma> says that if $g\in\mathbb Z_p[t]$, $g(t_0)\equiv0\pmod p$ and $g'(t_0)\not\equiv0\pmod p$, then there is a unique $t\in\mathbb Z_p$ with $t\equiv t_0\pmod p$ and $g(t)=0$. The point at infinity lifts to $O$. At an affine point $(\overline x,\overline y)$ of the reduced equation $F(x,y)=0$, being a <nonsingular point of an algebraic curve> means at least one <partial derivative> $F_x,F_y$ is nonzero. If $F_y\ne0$, fix any integral lift $x_0$ of $\overline x$ and apply the <Hensel lemma> to $g(t)=F(x_0,t)$ to lift $\overline y$. If $F_x\ne0$, fix $y_0$ and lift $\overline x$ instead. This proves the <surjectivity of good reduction over a local field> and gives
$$
\boxed{E(\mathbb Q_p)\twoheadrightarrow\widetilde E_p(\mathbb F_p)\text{ as abelian groups}.}
$$
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