For normal coworkers, it is good practice to let someone know the task is complete or they should ask. Your friend should take advantage of this situation and make himself look good.
As we discussed on Friday, I have taken on your task of XYZ while you were out in addition to my normal duties. I have gathered the data, created the TPS report and sent it to the clients (see attached). Hope you are feeling better.
I like that, and I will pass it along to him. Thanks.
My friend told me this morning that he was called into his boss's office, where his boss thanked him for maintaining progress on the tasks despite his coworker's absences. The boss told him that he has observed how the coworker tends to overreact and stress out. My friend said he thanked the boss for the compliment and reiterated his willingness to pitch in.
I told my friend that he may need to be mindful of the fine line between pitching in and doing his coworker's job for her. He said that if he believes he is doing her job for her, then he will take that to their boss. The e-mail documentation that has been suggested here will help to support that.
I also suggested that if/when the coworker's staff complains to him or his staff about their increasing workloads due to the coworker's absences, that's another signal that he needs to speak with his boss. In addition, if her staff is overloaded or running behind on tasks that affect his staff, then he should speak up about that as well. That's part of keeping his boss in the loop, and he can do that without complaining directly about the coworker's absences.
Here's hoping he keeps his spine strong!