We construct a generic polynomial for $\mathrm{Mod}_{2^{n+2}}$,
the modular 2-group of order $2^{n+2}$, with two parameters
over the $2^n$-th cyclotomic field $k$. Our construction is
based on an explicit answer for linear Noether's problem. This
polynomial, which has a remarkably simple expression, gives
every $\mathrm{Mod}_{2^{n+2}}$-extension $L/K$ with $K \supset k$,
$\sharp K = \infty$ by specialization of the parameters.
Moreover, we derive a new generic polynomial for the cyclic
group of order $2^{n+1}$ from our construction.