Solution:

A $31$-day month is $4$ weeks plus $3$ extra days, so exactly the weekdays of the $1$st, $2$nd, and $3$rd of the month occur five times.

For July to have five Mondays, Monday must be one of July $1,$ $2,$ or $3.$ In all three cases August $1$ lands on a Wednesday, Thursday, or Friday, and the common weekday among the resulting five-time days is Thursday.

More directly, since July has $31$ days, August $1$ is the same weekday as July $4.$ With Monday on July $1,$ $2,$ or $3,$ the three five-time weekdays of August always include Thursday.

Thus, the correct answer is D.