  In this article, we consider the so-called modified Euler scheme for
stochastic differential equations (SDEs) driven by fractional Brownian motions
(fBm) with Hurst parameter $\frac13<H<\frac12$. This is a first-order
time-discrete numerical approximation scheme, and has been recently introduced
by Hu, Liu and Nualart in order to generalize the classical Euler scheme for
It\^o SDEs to the case $H>\frac12$. The current contribution generalizes the
modified Euler scheme to the rough case $\frac13<H<\frac12$. Namely, we show a
convergence rate of order $n^{\frac12-2H}$ for the scheme, and we argue that
this rate is exact. We also derive a central limit theorem for the renormalized
error of the scheme, thanks to some new techniques for asymptotics of weighted
random sums. Our main idea is based on the following observation: the triple of
processes obtained by considering the fBm, the scheme process and the
normalized error process, can be lifted to a new rough path. In addition, the
H\"older norm of this new rough path has an estimate which is independent of
the step-size of the scheme.
