  In the current paper we study auto-B\"acklund transformations of the
non-stationary second Painlev\'e hierarchy $\text{P}_\text{II}^{(n)}$ depending
on $n$ parameters: a parameter $\alpha_n$ and times $t_1, \dots, t_{n-1}$.
Using generators $s^{(n)}$ and $r^{(n)}$ of these symmetries, we have
constructed an affine Weyl group $W^{(n)}$ and its extension $\tilde{W}^{(n)}$
associated with the $n$-th member considered hierarchy. We determined
$\text{P}_\text{II}^{(n)}$ rational solutions via Yablonskii-Vorobiev-type
polynomials $u_m^{(n)} (z)$. We brought out a correlation between
Yablonskii-Vorobiev-type polynomials and polynomial $\tau$-functions
$\tau_m^{(n)} (z)$ and found their determinant representation in the
Jacobi-Trudi form.
