Let the golfers be represented by the letters A to P.
Group 1: A,B,C,D
Group 2: E,F,G,H
Group 3: I,J,K,L
Group 4: M,N,O,P
The golfers in each group will automatically be considered to be co-players. Assuming each game consists of just two groups playing against one another, then, for example, the game between groups 1 and 2 (that is, group 1 versus group 2) will ensure that golfers A,B,C,D,E,F,G,H will be playing together.
There are 6 games in all to ensure that all the golfers play with (same group) or against (opposing group) one another, and each group will participate in three games:
Groups 1 versus 2, 1v3, 1v4, 2v3, 2v4, 3v4.
Group 1's games: 1v2, 1v3, 1v4
Group 2's games: 2v1, 2v3, 2c4
Group 3's games: 3v1, 3v2, 3v4
Group 4's games: 4v1, 4v2, 4v3
Therefore, 5 games will be insufficient to ensure that each golfer plays with each other golfer either by being in the same group or by being in an opposing group.