Let $G$ be a group. Recall that a group $H$ is called a retract of $G$ if there exist homomorphisms $g:G\longrightarrow{H}$ and $f:H\longrightarrow G$ so that $g\circ f=id_H$. The homomorphism $g$ is called a retraction. My question is that: Is there an infinite sequence $\{ H_i \}_{i\in \mathbb{N}}$ of groups so that the following conditions hold? 1- $H_1$ is a finitely presented group. 2- $H_{i+1}$ is retract of $H_i$ with the retraction $g_{i}:H_i \longrightarrow H_{i+1}$, for all $i\geq 1$; 3- $g_i$ is not an isomorphism for all $i\geq 1$. Thanks in advance.