The volitional form is often translated as "let's," but that's a first person plural imperative and I am not sure if that's the case for Japanese. In Japanese dictionaries, one of the definitions for the volitional form is:
勧誘や婉曲な命令を表す。下に「か」「じゃないか」などが付いて，意味を強めることがある。 「いっしょにジョギングでもしよう」 「少しおなかがすいてきた。すしでも食べようか」
At first glance, it seems that it's a second person imperative such as in expressions like 私と行こうよ. It would seem strange to order yourself to go somewhere with yourself. However, it being a second person imperative doesn't seem to work with other sentences such as みんなで行こうよ and 全員で行こうよ where "I" seems to be included with みんな and 全員. This leaves me at a loss as to who the 勧誘や婉曲な命令 is directed to when the subject is omitted and it's just a verb. For example, if a group of people were about to go somewhere and someone said 行こう, is it a second person imperative with 私と omitted or a first person plural imperative?