FA20 AND LEARJET WERE CONFUSED BY GND CTLRS AT OAK.

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NOV/THU/01 AT OAK ARPT AT APPROX XA00; TAXIING FROM FBO TO RWY 29; THERE HAPPENED TO BE A LEARJET THAT HAD TAXIED OUT BEFORE US AND WAS XING RWY 27R BEFORE WE EVEN LEFT THE PARKING RAMP. TAXI INSTRUCTION TO US WAS TO TAXI E TO TXWY B AND HOLD SHORT OF RWY 27R. WE READ BACK OUR TAXI INSTRUCTION AND COMPLIED WITH THEM. AS WE APCHED RWY 27R; WE WERE TOLD TO PROCEED ACROSS RWY 27R AND HOLD SHORT AT RWY 27L; WHICH WE READ BACK. WE PROCEEDED ACROSS RWY 27R ON TXWY B AND WERE HOLDING SHORT AT RWY 27L. THE LEARJET MEANWHILE HAD TAXIED ON TXWY B AND WAS HOLDING AT TXWY B1 WAITING FURTHER INSTRUCTIONS. OUR GND CTLR AT THIS POINT GOT US CONFUSED WITH THE LEARJET AND TOLD US TO CONTACT ANOTHER GND CTL FREQ; WHICH WE DID. WE CALLED THE SECOND GND CTLR AND ADVISED WE WERE HOLDING SHORT AT RWY 27L ON TXWY B. SHE THEN ASKED IF WE WERE ABEAM THE HANGARS ON TXWY B. WE WERE LOOKING AT THE HANGAR STRAIGHT IN FRONT OF US WITH THE SIDE OF THE HANGAR TOWARDS US. THEREFORE; WE THOUGHT HER REF TO THE HANGAR WAS TO US ABEAM WHICH WE REPLIED AGAIN THAT WE HOLDING SHORT RWY 27L ABEAM THE HANGARS. MEANWHILE; SHE THOUGHT WE WERE THE LEARJET HOLDING SHORT AT TXWY B1; WHICH WAS ACTUALLY IN FRONT OF THE HANGAR WAITING FOR FURTHER INSTRUCTION TO TAXI. SHE THEN TOLD US TO TAXI TXWY B AND TXWY W TO RWY 29. AGAIN; WE READ BACK THE INSTRUCTION AND CHKED FOR TFC BEFORE XING RWY 27L. WE SAW A C170 ON BASE LEG TURNING FINAL. WE KNEW WE HAD PLENTY OF TIME TO CROSS RWY 27L. MEANWHILE THE LEARJET HOLDING AT TXWY B1 WAS GETTING A LITTLE IRRITATED BY THE SOUND OF HIS VOICE; BECAUSE THE GND CTLR WAS NOT RECOGNIZING HIS CALL. SHE FINALLY ANSWERED HIM AND TOLD HIM TO TAXI TO RWY 29. MEANWHILE; AS WE TAXIED UP TO TXWY B1 WHERE THE LEARJET WAS JUST STARTING TO ROLL; WE ASKED HER IF SHE WANTED US TO 'CONTINUE TO FOLLOW THE LEARJET OR SHOULD WE HOLD AT TXWY B1?' SHE THEN REPLIED; 'I THOUGHT YOU WERE ABEAM HANGAR BEFORE.' WE TOLD HER WE WERE HOLDING SHORT OF RWY 27L ON TXWY B. THERE WAS SILENCE AND SHE TOLD US TO CONTINUE TO RWY 29 BY TXWY B AND TXWY W. AGAIN WE READ BACK THE INSTRUCTION. AS WE APCHED RWY 29 AND WERE CLRED INTO POS; WE WERE ASKED BY TWR AND GIVEN A PHONE NUMBER TO CALL. WHEN WE ARRIVED AT LGB; WE GAVE OAK TWR A CALL. THE PERSON THAT ANSWERED OUR CALL KNEW NOTHING ABOUT WHAT HAD HAPPENED; BUT AS I EXPLAINED WHAT WENT ON; HE INFORMED US THE TWR CHIEF/SUPVR WAS DOWNSTAIRS LISTENING TO THE TAPES. HE THEN TOOK OUR PHONE NUMBER AND SAID THE TWR CHIEF WOULD GIVE US A CALL ONCE HE RETURNED. MEANWHILE; HE EXPLAINED TO US THAT AT OAK WE SHOULD HAVE STAYED WITH THE N GND CTLR FREQ UNTIL REACHING TXWY B1 AT MIDFIELD. HE ALSO ASKED WHEN WAS THE LAST TIME WE HAD BEEN IN OAK. WHICH TO ME WAS AN UNUSUAL QUESTION; BUT I TOLD HIM THAT I HADN'T BEEN THERE FOR 4 OR 5 YRS. HE ASKED IF WE WERE SURE THAT WE HADN'T BEEN THERE LAST WK; WHICH WE REPLIED NO. SHORTLY THEREAFTER; THE TWR CHIEF/SUPVR CALLED US AND EXPLAINED WHAT HAD HAPPENED. FIRST HE STATED THAT THE FIRST CTLR HAD GOTTEN US MIXED UP WITH THE LEARJET AND APPARENTLY AFTER GIVING US A FREQ CHANGE; THINKING WE WERE THE LEARJET; TRIED TO CONTACT US TO CROSS RWY 27L. REALIZING THAT WE WERE NOT ON HIS FREQ ANYMORE AND HAD NOT READ BACK TAXI INSTRUCTION TO HIM; THE OTHER GND CTLR THOUGHT WE WERE THE LEARJET AND THAT IS WHY SHE TOLD US TO TAXI BY TXWY B TO TXWY W TO RWY 29. NOT BEING FAMILIAR WITH THE PROC AT OAK; WE THOUGHT WE WERE FOLLOWING INSTRUCTIONS AS THEY WERE GIVEN TO US. WE READ BACK THE TAXI INSTRUCTION AS ALWAYS; BUT THE CTLRS BECAUSE OF THE HVY WORKLOADS; APPARENTLY DID NOT COMPREHEND WHAT WE HAD SAID. HOWEVER; DUE TO THE FACT WE ALWAYS CHK BOTH WAYS BEFORE XING A RWY; NO CLOSE CALL WAS ENCOUNTERED AND WE HAD THE CESSNA IN SIGHT. I QUESTIONED THE TWR CHIEF/SUPVR IN REGARDS TO WHAT ACTION WOULD BE TAKEN. HE STATED THAT HE WOULD CONSIDER THIS A LEARNING SIT FOR THE CTLRS AS WELL AS FOR US. HE ALSO INFORMED US THAT THE FIRST GND CTLR HAD CLRED US TO CROSS RWY 27L; EVEN THOUGH WE WERE NOT ON HIS FREQ AND THEREFORE HAD NOT READ BACK HIS INSTRUCTIONS. THE TWR CHIEF AGAIN WENT OVER THE PROCS USED AT OAK FOR CONTACTING GND CTL AND WHERE AND WHEN TO SWITCH FREQS. THE TWR CHIEF/SUPVR CONSIDERED THIS MATTER CLOSED AND SO DO I. HOWEVER; I HOPE OTHER PLTS MAY LEARN FROM THIS EXPERIENCE; NOT ONLY TO BE ALERT XING A RWY; BUT IF THE GND CTLRS APPEAR TO BE CONFUSED AS TO WHICH PLANE THEY ARE HANDLING; FOR THE PLT TO QUESTION ALL INSTRUCTIONS BEFORE PROCEEDING TO TAXI. ALSO; TO CHK BOTH DIRECTIONS BEFORE XING ANY RWY; WHICH WE DID; SO THAT A MISTAKE LIKE THIS DID NOT PUT US IN HARM'S WAY.