These Days Work for Me
Perl 6, 47 bytes



{my$d=Date.new($_);map($d+*-$d.day-of-week,^7)}


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The code is almost self-explanatory. First we make up a Date object $d from the string, which happens to be in the nice ISOwhatever format. We can perform arithmetic with the dates (adding an integer = adding that many days, the same with subtracting). So $d-$d.day-of-week is the last this (oh my god, week has always started with Monday for me :D) week's Sunday. (.day-of-week is 1 for Monday to 7 for Sunday.) We then map over 0 to 6 (^7) and add that to the Sunday, getting all the days in the week. The number we're mapping over appears in the place of the star *. This list is then returned.