ZHU controller described an event involving an aircraft being transferred to another sector with inaccurate altitude assignment; the reporter alleging an incomplete relief briefing was the causal factor in this instance.

Narrative

Working positions 85 and 50 combined with multiple frequencies. [I was] working approximately 17 aircraft with military aircraft in west MOAS; also working weather deviations within the sector. [I] received a relief briefing from trainee and OJTI (on job training instructor). Got sufficient briefing on all aircraft; Aircraft X was briefed climbing to FL190. This information was wrong aircraft was assigned 170 by trainee. When I took over sector it was very busy with over 4 frequencies in use causing a lot of congestion. Approximately 3 minutes later a T38 in west MOA declared emergency. Also at this time; D-Side was giving a relief briefing to get out of position. At this point my attention went to emergency and asking D-Side for help and working all other aircraft. All I had done with Aircraft X was give him a frequency change to Forth Worth Center; because I was told he was climbing to FL190. Unknown to me until after an investigation was done; was that Forth Worth Center tried to call; to ask why Aircraft X was stopped at 170; due to emergency and whether the D-side or myself answered Forth Worth Sector F62? I did not know of event until later; I was notified that I had switched aircraft to Forth Worth at XA30z. I'm not sure how I'm involved in this for just switching aircraft to Forth Worth I was also told he was climbing to FL 190 hard altitude. Recommendation; get better briefing.