Lawrence did not empower the Hashemites, the House of Hussein. The Hashemites were considered the direct descendants of Mohammed and were the appointed rulers of Mecca and Medina under the Turkish Caliphate.
In 1924 they were overthrown by Ibn Saud who allied with the Inkwan, translated as The Brotherhood. This was an alliance of convenience, and Ibn Saud defeated the Inkwan but kept Wahabbi rule.
I'm not sure what if any relationship the Inkwan has to the Muslim Brotherhood.