Solution of @spender is good, but can it be done without OrderBy? It can, because we can use the index for direct indexing into array, but it would not be one linq query:
var resultsTmp =
DateList.Select((d, i) => new { d, i })
.GroupBy(x => x.d.Date.Subtract(firstDay).Days / 7 + 1)
.SelectMany(gx => gx, (gx, x) => new { Week = gx.Key, DateTime = x.d, Count = gx.Count(), x.i })
.ToArray();
var resultsTmp2 = resultsTmp.ToArray();
foreach (var r in resultsTmp) { resultsTmp2[r.i] = r; };
var results = resultsTmp2.Select(r => new { r.Week, r.DateTime, r.Count });
It looks a bit complex. I would probably do something more straightforward like:
var DateList2 = DateList.Select(d => new { DateTime = d, Week = d.Subtract(firstDay).Days / 7 + 1 }).ToArray();
var weeks = DateList2.GroupBy(d => d.Week).ToDictionary(k => k.Key, v => v.Count());
var results = DateList2.Select(d2 => new { d2.Week, d2.DateTime, Count = weeks[d2.Week] });