The function $\sigma:x\to x^{-1}$ induces a function on the ${\rm C}^*$-algebra ${\mathbb C}H$ that sends positive elements to positive elements; indeed, this function is akin to the usual involution on the group ${\rm C}^*$-algebra except that it is linear rather than conjugate linear. Consequently, composing any homomorphism $f:G\to H$ with $\sigma$ yields a function $\sigma f$ whose extension to ${\mathbb C} G\to {\mathbb C}H$ sends positive elements to positive elements, yet which will usuall ynot be a homomorphism (unles $H$ is abelian).

This example suggests replacing the positivity property in your question by "complete positivity", but I have not thought about whether this stronger condition is enough to force your function $f$ to be a homomorphism.

The function $\sigma:x\to x^{-1}$ induces a function on the ${\rm C}^*$-algebra ${\mathbb C}H$ that sends positive elements to positive elements; indeed, this function is akin to the usual involution on the group ${\rm C}^*$-algebra except that it is linear rather than conjugate linear. Consequently, composing any homomorphism $f:G\to H$ with $\sigma$ yields a function $\sigma f$ whose extension to ${\mathbb C} G\to {\mathbb C}H$ sends positive elements to positive elements, yet which will usually not be a homomorphism (unless $H$ is abelian).

P.S. This example suggests replacing the positivity property in your question by "complete positivity", but I haven't thought this through.