Arjuna begins the fifth chapter by asking Krishna to stop praising two things at once. You commend renouncing actions, he says, and you also commend disciplined action; tell me definitely which one is better. It is the complaint of someone trying to get a straight instruction out of a teacher who keeps widening the frame.
The reply concedes that both lead to the highest good and then declines the choice anyway: of the two, action performed with discipline is the better. The reason given is not that renunciation is wrong but that it is harder to reach without the other. Renunciation is difficult to attain without disciplined action, and the person who is disciplined in action comes to the goal before long.
What the chapter then does is dissolve the distinction it was asked to adjudicate. The true renouncer is defined as the one who neither hates nor craves, and such a person is free whether or not they have left their household. Someone who acts having given over their actions and abandoned attachment is untouched by wrongdoing as a lotus leaf is untouched by water. The body, the senses and the mind go on working, but the doer has stopped claiming the work.
The chapter also contains the image of the embodied self dwelling at ease in a city of nine gates, neither acting nor causing action, and the reminder that it is nature that acts throughout. Read alongside chapter three, this is the Gita's consistent position: the argument is never about whether to be busy or idle, but about whether the person doing the work believes themselves to be its owner.