Is it permissible to kill a man who sends messages containing pictures of another man's wife, claiming to be in a relationship with her, this being an act of defending one's honor?

It is not permissible to accuse a man of sending messages or having a relationship with a woman without proof, because the default assumption for Muslims is innocence. If these actions are proven to have originated from him, then he must be advised and reminded of Allah, and he can be threatened with escalating the matter to someone who will discipline him. However, killing is not permissible in this situation under the pretext of defending one's honor.