Moodle can be set up for assignments that only require one member of a group to submit a document. Grading the assignment will send the same grade to all members of the group.
Create a Moodle assignment as usual, but adjust the settings for group submission.
Under "Group submission settings" section
Under "Common Module settings" section
Students will submit files to the assignment as usual, but when one group member submits a file, it will show up under all the group members' submissions. Grading one group member will send the grade to all group members.
It is more difficult to set up assignments if the groups change throughout the semester, but it is possible using something called a "Grouping". Groupings are sets of groups.
Create a "Week 1" Grouping and then create and populate Groups A, B, and C within the Week 1 grouping. Create the assignment for Week 1 and specify the "Week 1" grouping in the Group Submission Settings → Grouping for student groups.
Repeat the same process for Week 2 with a new grouping and new groups A, B, and C within the Week 2 grouping.
It can get time consuming to create the groups. It is possible to import a csv file with the group memberships, if that is an easier format to generate: https://docs.moodle.org/37/en/Import_groups