collision resistance of a composed hash function Given a collision-resistant hash function $H(x)$, say SHA256, and I devise another hash function by $H'(x)=\sum^{m}_{1}H(x||i) \mod p$. The summation is defined on finite field of prime order $p$. 

Is the new hash function $H'(x)$ still collision resistant? How hard is it to find a collision? Any advice would be greatly appreciated!
Counterexample [from poncho in comments]: let $H_0\colon \{0,1\}^* \to \{0,1\}^{256}$; then $H(x) := p \cdot H_0(x)$, where $H_0(x)$ is interpreted as an integer in little-endian, is obviously collision-resistant but $H'(x) := \sum H(\cdots) \bmod p$ is identically zero and therefore very much not collision-resistant no matter what goes in the ellipsis.