The density of odd order reductions for elliptic curves with a rational point of order 2

Suppose that $E\mathrm{/}\mathbb{Q}$ is an elliptic curve with a rational point $T$ of order $2$ and $\alpha \in E(\mathbb{Q})$ is a point of infinite order. We consider the problem of determining the density of primes $p$ for which $\alpha \in E($\mathbb{F}${p})$ has odd order. This density is determined by the image of the arboreal Galois representation $\tau{E,2^{k}} : {\rm Gal}(\overline{$\mathbb{Q}$}/$\mathbb{Q}$) \to {\rm AGL}_{2}($\mathbb{Z}$/2^{k}$\mathbb{Z}$)$\mathrm{.}Assumingthat$\alpha$ is primitive (that is, neither $\alpha$ nor $\alpha +T$ is twice a point over $\mathbb{Q}$) and that the image of the ordinary mod $2^k$ Galois representation is as large as possible (subject to $E$ having a rational point of order $2$), we determine that there are $63$ possibilities for the image of ${\tau }_{E,2^k}$. As a consequence, the density of primes $p$ for which the order of $\alpha$ is odd is between $1\mathrm{/}14$ and $89\mathrm{/}168$.