  Let $K$ be an algebraically closed field of characteristic $2$, $G$ be the
algebraic group $\mathrm{SL}_2$ over $K$, and $V$ be the natural representation
of $G$. Let $b_k^{G,V}$ denote the number of $G$-indecomposable factors of
$V^{\otimes k}$, counted with multiplicity, and let $\delta = \frac 32 -
\frac{\log 3}{2\log 2}$. Then there exists a smooth multiplicatively periodic
function $\omega(x)$ such that $b_{2k}^{G,V} = b_{2k+1}^{G,V}$ is asymptotic to
$\omega(k) k^{-\delta}4^k$. We also prove a lower bound of the form $c_W
k^{-\delta}(\dim W)^k$ for $b_k^{G,W} $ for any tilting representation $W$ of
$G$.
