Automate On Call Schedule

Hi,

I wasn't sure if this should be posted under general or in the formula section. Sorry if I've posted it in the incorrect place.

At my job, I have to create a schedule for crews that will be on call and crews that will be working Saturday. I've attached a spreadsheet with some generic names to show what I'm trying to do. In the attached spreadsheet, I've figured out a way to automate columns A - G. I cannot figure out a function that will automate column H. I was hoping that someone could help me come up with a function to automate this column. Here are the basic rules to this column:

1) If someone is on call Monday, they cannot be the Saturday crew (column H) for that week.
2) If someone is on call Fri - Sun, they cannot be the Saturday crew (column H) for the week.
3) We want to try to balance the number of Saturdays each crew works so it is as even as possible.

I don't know if this helps at all, but the rotation of columns E-F goes as follows:

Mon -> Fri - Sun -> Thurs -> Wed - Tue -> Mon (repeat)

Any help is appreciated. I've been killing my brain trying to figure this out.

Thanks in advance.