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In this paper, we study some relation between the cocenter $\bar H(G)$ of theHecke algebra $H(G)$ of a connected reductive group $G$ over an nonarchimedeanlocal field and the cocenter $\bar H(M)$ of its Levi subgroups $M$.  Given any Newton component of $\bar H(G)$, we construct the induction map$\bar i$ from the corresponding Newton component of $\bar H(M)$ to it. We showthat this map is surjective. This leads to the Bernstein-Lusztig typepresentation of the cocenter $\bar H(G)$, which generalizes the work \cite{HN2}on the affine Hecke algebras. We also show that the map $\bar i$ we constructedis adjoint to the Jacquet functor and in characteristic $0$, the map $\bar i$is an isomorphism.

Cocenter of $p$-adic groups, II: induction map