A WRONG DIRECTION OF TURN IS MADE TO THE ASSIGNED INITIAL VECTOR HEADING OF 050 DEGS DURING A DEP FROM RWY 23 AT FWA.

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ACFT X WAS HOLDING SHORT OF RWY 23 HAVING ALREADY COMPLETED THE BEFORE TKOF CHK 'TO THE BOX;' AS PER COMPANY PROFILE. THE PNF ADVISED THE CTL TWR THAT THEY WERE READY FOR DEP. FWA TWR CLRED ACFT X FOR DEP WITH A CLRNC STATING THE CURRENT WINDS; POSSIBLY THE ALTIMETER SETTING; A NEW INITIAL ALT OF 10000 FT FROM THE ORIGINAL 5000 FT; AND A HEADING OF 050 DEGS TO JOIN THE MIZAR 3 ARR INTO DETROIT. THE PNF READ BACK THE CLRNC; COMPLETED THE 'BOXED' ITEMS OF THE BEFORE TKOF CHK; SET THE NEW ALT; AND BUGGED THE HEADING OF 050 DEGS. AT THE SAME TIME; THE PF TAXIED THE ACFT ONTO THE RWY TO BEGIN A ROLLING TKOF. NORMAL PROCS WERE FOLLOWED FOR THE TKOF. AFTER ROTATION; ACFT X'S TKOF PROFILE CALLS FOR SPD MODE (FROM THE AUTOMATIC FLT CTL SYS THAT DIRECTS/COMMANDS THE FLT DIRECTOR) TO BE SELECTED AT V2 +10 KTS WHICH THE PF CALLED FOR AND THAT MODE WAS SELECTED BY THE PNF. AT 400 FT AGL AS PER PROFILE; THE PF CALLED FOR HEADING MODE FROM THE AFCS AND THE PNF SELECTED HEADING MODE. AT THIS TIME; THE FLT DIRECTOR COMMAND BARS INITIATED A L TURN TO THE PRESET BUG OF 050 DEGS. THE PF CLRED THE AIRSPACE TO THE L AND BEGAN THE TURN TO 050 DEGS TO JOIN THE MIZAR 3 ARR. AT THAT TIME; THE PF QUERIED THE PNF IF THE INITIAL TURN WAS TO BE L OR R. NEITHER CREW MEMBER COULD RECALL CORRECTLY AND BEFORE THE PNF WAS ABLE TO VERIFY; FWA TWR XMITTED; 'ACFT X; YOUR INITIAL TURN WAS TO BE A R TURN TO 050 DEGS TO JOIN THE MIZAR 3; BUT NO PROB; CONTINUE L AND CONTACT DEP.' THE PF CONTINUED THE L TURN TO 050 DEGS AND THE PNF CONTACTED DEP CTL. AT THIS TIME; THE DEP CTLR ASKED WHY WE DEVIATED FROM OUR ORIGINAL TKOF CLRNC. THE PNF EXPLAINED THAT; 'WHEN THE HEADING MODE WAS SELECTED; THE COMMAND BARS INITIATED A L TURN.' SINCE THE HEADING IN THE TKOF CLRNC WAS 180 DEGS FROM RWY HEADING; THE AFCS HEADING MODE INITIATED ITS OWN PREDETERMINED COURSE OF ACTION. FWA DEP CTL EXPLAINED THAT IF THIS SIT OCCURRED IN THE AFTERNOON; WE WOULD HAVE TURNED INTO AN AREA OF PRACTICING F16'S. DEP SAID THAT WE WERE NOT THE ONLY ONES THAT HAD MADE THE L TURN AND IT WAS BECOMING A REOCCURRING PROB. DEP THEN AGAIN ASKED WHY WE HAD MADE A L TURN INSTEAD OF A R TURN. THE CREW GAVE THE SAME EXPLANATION. WE WERE CLRED TO JOIN THE MIZAR 3 ARR WITH ANOTHER HEADING FROM DEP CTL. THE ISSUE IS WHETHER OR NOT THE PNF READ BACK THE CORRECT DIRECTION; OR ANY DIRECTION OF TURN FOR THAT MATTER; FOR THE 050 DEGS TO JOIN THE MIZAR ARR. THE CREW CANNOT POSITIVELY RECALL IF THEY HEARD R TURN OR NOT -- MERELY JUST A 050 DEG HEADING. TWR HAD THE ADEQUATE VISIBILITY TO IMMEDIATELY CORRECT THE CREW FROM THE L TURN WE HAD INITIATED. FATIGUE MAY HAVE ALSO PLAYED A SMALL ROLE AS THE CREW HAD ARRIVED AT THE HOTEL THE NIGHT BEFORE AT APPROX XA45 LCL; DUE TO A DELAY IN HOTEL TRANSPORTATION; AND HAD AN XA00 VAN RIDE TO THE ARPT.