Can't we just take $G$ to be the union of the chain $H_1<H_2<\cdots$, where $H_1$ is a Klein 4-group and $H_i$ is the dihedral group of order $2^{i+1}$? Then $N_{G}(H_i) = H_{i+1}$.

Can't we just take $G$ to be the union of the chain $H_1<H_2<\cdots$, where $H_1$ is a Klein 4-group and $H_i$ is the dihedral group of order $2^{i+1}$? Then $N_{G}(H_i) = H_{i+1}$.

So

$G = \langle x_i (i \ge 0), y \mid x_1^2=1, x_{i+1}^2=x_i (i\ge 1), y^2=1, (yx_i)^2 = 1 (i \ge 1) \rangle$,

where $H_i$ is the subgroup $\langle x_i, y \rangle$.