Flight Instructor reported they observed their solo student take off from the non-towered airport; enter downwind and have a NMAC with an aircraft entering the pattern.

Narrative

I was supervising instructor of a student pilots second solo. The student had just taken off runway XX when another aircraft announced they were inbound from the north to land on runway XX. My student had announced they were turning crosswind and the other aircraft stated they did not have the departing aircraft in sight. My student misheard and believed the other aircraft had them in sight. Consequently as my student was nearing their turn to downwind the other aircraft was already in the downwind and the two aircraft nearly collided. My student made a sharp turn to the left and the other aircraft made a sharp turn to the right and rejoined the pattern behind my student. My student landed without incident and came back to park. I believe miscommunication and poor situational awareness by both parties led to this situation as well as nonstandard pattern entry by the other aircraft. To avoid this situation; it is important to know where arriving aircraft are located as well as know their intentions. It's also important to communicate clearly so that there is no confusion. To correct this situation I believe my student could have extended their departure leg until they had a visual on the other aircraft; my student could have also radioed the other aircraft to ask for a position update before turning. I also believe if the other aircraft had remained clear of the pattern until my student was in the downwind leg rather than entering directly into a downwind that a near miss would have been avoided.