  The mod $2$ Lannes-Zarati homomorphism was constructed in \cite{Lan-Zar87},
which is considered as a graded associated version of the mod $2$ Hurewicz map
in the $E_2$-term of Adams spectral sequence. The map is studied by many
authors such as Lannes-Zarati \cite{Lan-Zar87}, H\horn{u}ng \cite{Hung97},
\cite{Hung2001}, \cite{Hung2003}, H\horn{u}ng et. al. \cite{Hung.et.al2014},
Ch\horn{o}n-Tri\'\^et \cite{Chon.Triet2014}. In this paper, we construct an
analogue $\varphi_s$ for $p$ odd, and we also investigate the behavior of this
map for $s\leq 3$.
