Dame Elish Angiolini’s report called for a restriction on previous police policemans leading protection death examinations.
Ms Sturgeon stated priests would certainly “thoroughly think about” the report, yet said there were differences between the English as well as Scottish system for wardship plans.
Ms Baker added: “I’m let down in the action from the First Minister.
She also wanted a team of independent investigators to be on call 24 hours a day.
Mr Bayoh, 31, died after being restrained by officers responding to a call in Kirkcaldy in May 2015.
Speaking at First Minister’s Questions, Claire Baker, a Labour MSP in Fife, said: “Will the First Minister today commit to holding an inquiry and can I also ask for her response to Dame Elish Angiolini’s report?”
Ms Sturgeon said ministers would “carefully consider” the report, but said there were differences between the English and Scottish system for custody arrangements.
Ms Baker added: “I’m disappointed in the response from the First Minister.
“It is vital Sheku Bayoh’s family get answers over what happened the day he died.”