  The $2$-primary homotopy $\beta$-family, defined as the collection of
Mahowald invariants of Mahowald invariants of $2^i$, $i \geq 1$, is an infinite
collection of periodic elements in the stable homotopy groups of spheres. In
this paper, we calculate $\mathit{tmf}$-based approximations to this family.
Our calculations combine an analysis of the Atiyah-Hirzebruch spectral sequence
for the Tate construction of $\mathit{tmf}$ with trivial $C_2$-action and
Behrens' filtered Mahowald invariant machinery.
