= Solution
At $2$ there is <good reduction>, and the <surjectivity of good reduction over a local field> gives an <exact sequence>
$$
0\longrightarrow E_1(\mathbb Q_2)\longrightarrow E(\mathbb Q_2)\longrightarrow\widetilde E_2(\mathbb F_2)\longrightarrow0.
$$
The quotient has order $5$ and is consequently a <cyclic group>. For every odd integer $n$, multiplication by $n$ is an automorphism of $E_1(\mathbb Q_2)$: its <formal group law> has multiplication series $nt+O(t^2)$ with unit linear coefficient, whose inverse <formal power series> has integral coefficients and converges on $2\mathbb Z_2$. This is the <prime-to-residue-characteristic multiplication on a formal group>.
Thus the odd-primary <torsion subgroup> injects into the order-five quotient. It also realizes that quotient: choose a lift $P$ of any reduced point, so $[5]P\in E_1$. There is a unique $Q\in E_1$ with $[5]Q=[5]P$. Then $P-Q$ is killed by five and has the prescribed reduction. This proves the <prime-to-p torsion lifts uniquely at good reduction> directly here. Therefore, for an odd prime $p$,
$$
\boxed{E(\mathbb Q_2)[p^\infty]\cong\begin{cases}\mathbb Z/5\mathbb Z,&p=5,\\0,&p\ne5.\end{cases}}
$$
The five-primary subgroup consists of the unique five-torsion lifts of the five reduced points. These lifts need not be rational over $\mathbb Q$; the preceding answer about rational <torsion points of an elliptic curve> does not contradict their existence over the <p-adic field>.
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