Bousfield p-completion on spaces is a functor $(-)^{\wedge p}$ whose main property is that a map $f:X\rightarrow Y$ induces an isomorphism $f_{\ast}:H_\ast(X,\mathbb{F}_{p})\rightarrow H_\ast(Y,\mathbb{F}_p)$ if and only if $f^{\wedge p}:X^{\wedge p}\rightarrow Y^{\wedge p}$ is a homotopy equivalence.

Let $X$ be now be a connective CW spectra, it is known that $H\mathbb{F}_p$-localization agrees with $M\mathbb{F}_p$-localization (both on X, the latter is referred as the $p$-completion). Let $f:E\rightarrow F$ a map between two connective CW spectra, is there any connection between $f^{\wedge p}$ and $({H\mathbb{F}_p})_{\ast}(f)$ (or $({H\mathbb{F}_p})^{\ast}(f)$)?, of course, an analogous version of the previous property is desirable, although there may be more restrictive conditions so that this property holds.