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Google Calendar is good method for sharing schedule.
Surely, user can input directly it. But manager needs another form.
So, manager can receive members working list with date specific form.
The specific form made by Google Form or Google Spreadsheet, like below:
After members report their work data, manager handle it freely.
I recommend Zapier to you, like this.
If your setting has no problem, Google Calendar says "members schedule", easily.
Zapier will support you to write required fileds, kindly.
Go!