Solution:

Since July has $31=4\cdot7+3$ days, the weekday of July $1$ occurs five times, so Monday falls on July $1,$ $2,$ or $3.$ The days that occur five times in August are those of August $1,$ $2,$ $3,$ and August $1$ is three weekdays after July $1.$

Testing the three cases, the August weekdays occurring five times are $\{\text{Thu},\text{Fri},\text{Sat}\},$ $\{\text{Wed},\text{Thu},\text{Fri}\},$ and $\{\text{Tue},\text{Wed},\text{Thu}\}.$ Thursday appears in every case.

Thus, the correct answer is D.