A note on multiplicative functions resembling the M\"obius function

We provide examples of multiplicative functions $f$ supported on the squarefree integers, such that on primes $f(p)=\pm 1$ and such that $M_f(x):={\sum }_{n\le x}f(n)=o(\sqrt{x})$. Further, by assuming the Riemann hypothesis (RH) we can go beyond $\sqrt{x}$-cancellation.