Consider the group $\displaystyle G=\mathbb{Z}_2\ast\mathbb{Z}_3=<x,y:x^2=y^3=e>$ and consider the surjective homomorphisms $\displaystyle G\twoheadrightarrow\mathbb{Z}_2$, $\displaystyle G\twoheadrightarrow\mathbb{Z}_3$ and $\displaystyle G\twoheadrightarrow\mathbb{Z}_6$. Find a presentation and the abelianization of each kernel.