The man has been caught on surveillance video targeting women in broad daylight.
Video captured during one incident shows the man dragging an 81-year-old woman in an attempt to steal her purse.
The woman, who did not want to be identified, says she suffered a broken arm, and will soon undergo surgery.
"We don't want you to get hurt. If they want your purse, give it to them. Don't put up a fight," HPD officer Don Von said.
On Thursday, authorities released a sketch of the suspect.
Investigators say the man is accused of snatching at least five other women's purses in the Memorial and Sugar Land area.
"We have to be alert and aware of our surroundings because criminals target those that are the most vulnerable and that aren't paying attention," Von said.
If you're out shopping this holiday season, you may want to make sure your shopping bags and purse are secure by using the child safety belt buckle on shopping carts.
"You can lock your purse in place, and that way if someone tries to come up and grab it, they can't," Von said.
Also, you want to double check that you are not being followed once you leave the store. If you are being followed, make sure to stay in well lit areas or ask an officer or security guard to escort you from the store.
"Tap them on the shoulder and let them know, 'Hey, I'm walking to my car and I feel uneasy. Do you mind watching me as I go to my car?'" Von said.
HPD also says the best practice is to shop with a partner, and do not leave bags and expensive things in your car for long.