  We study the average error term in the usual approximation to the number of
$y$-friable integers congruent to $a$ modulo $q$, where $a\neq 0$ is a fixed
integer. We show that in the range $\exp\{(\log\log x)^{5/3+\varepsilon}\} \leq
y \leq x$ and on average over $q\leq x/M$ with $M\rightarrow \infty$ of
moderate size, this average error term is asymptotic to $-|a|\Psi(x/|a|,y)/2x$.
Previous results of this sort were obtained by the second author for reasonably
dense sequences, however the sequence of $y$-friable integers studied in the
current paper is thin, and required the use of different techniques, which are
specific to friable integers.
