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Hi, Felipe. I actually think you are wrong now. There are always elements in Z/pZ has order 2 since the multiplicative group has order p-1 which is an even number if p is odd.
You can always use a nontrivial bundle over a sphere one point join with a trivial bundle over a space satisfying those properties to construct such example.
Agol, Can you tell me why when it is a HNN extension, the answer should be yes? As a matter of fact, my group arise from some topology construction, if I calculated the fundamental group right, it is an HNN extension.