(written from a Production point of view)
The Star Trek II: The Wrath of Khan Game is one of Tsukuda Hobby's SF Series games (item no. HG-017-3500, the last four digits indicating the price in Japanese yen). Produced in 1983 by the Japanese toy/game maker, and designed by Kazutoyo Ishii, it was only released in Japan, while bearing the additional subtitle "The Game of Space War in Star Trek".
This is a strategic hex- and counter-based ship-to-ship combat game based on the second Star Trek movie. It seems to be simply an upgraded version of their previous Star Trek: The Invasion of Klingon Empire game, but limiting the sides to just the Federation and Klingon Empire. However, the combat would look to be expanded to allow ships to separate into what's known as "booms and saucers", meaning the ships can separate their command sections from their "stardrive" sections and still battle on. (Like its predecessor, this type of ship combat is again inspired by similar possible game play in the Star Fleet Battles series of games.)
It claims two to four players, but like many of Tsukada's games (including Invasion), you have to split into two "teams" and share counters for more than two players; it is really just a two-player game. The game includes eight different scenarios to fight, including a recreation of the featured battle in the movie between the USS Enterprise and the USS Reliant.