Let $\varphi_i:G\to H_i$ for $i=1,2$ be two group homomorphisms. I want to find elements in $\mathrm{Kern}(\varphi_1)\cap \mathrm{Kern}(\varphi_2)$ which are not contained in the commutator $[G,G]$. Is there any systematic way to do this?
In my special case, I am in the following situation: Let $G=\langle a_1,\dotsc,a_n\rangle$ be a free group and let $H=\langle{a_1,\dotsc,a_{r}}\rangle$ with $r<n$. Now consider words $v_{r+1},\dotsc,v_n\in H$ and $w_{r+1},\dotsc,w_n\in H$ and the corresponding two homomorphisms $\pi_1,\pi_2:G\to H$ sending $a_i$ to $a_i$ for $i\le r$ and $a_i\mapsto v_i$ resp. $a_i\mapsto w_i$ for $i\ge r+1$. Can I find an $[G,G]\not\ni g\in \mathrm{Ker}(\pi_1)\cap \mathrm{Ker}(\pi_2)$?