  We provide examples of multiplicative functions $f$ supported on the $k$-free
integers such that at primes $f(p)=\pm 1$ and such that the partial sums of $f$
up to $x$ are $o(x^{1/k})$. Further, if we assume the Generalized Riemann
Hypothesis, then we can improve the exponent $1/k$: There are examples such
that the partial sums up to $x$ are $o(x^{1/(k+\frac{1}{2})+\epsilon})$, for
all $\epsilon>0$. This generalizes to the $k$-free integers the results of
Aymone, `` A note on multiplicative functions resembling the {M}\"{o}bius
function'', J. Number Theory, 212 (2020), pp. 113--121.
