The groups said the CID boss is "incompetent and insensitive" in relation to her public comments on the Takoradi kidnapped girls.
Ruth Quayson, Priscilla Blessing Bentum and Priscilla Koranchie were reportedly kidnapped between August 2018 and January 2019.
In April, the Police CID boss at a press conference disclosed that they had discovered the whereabouts of the girls.
COP Maame Yaa Tiwaa Addo-Danquah said the police was putting measures in place to hand them to their families.
COP Addo-Danquah said: "We know where the girls are."
However, nothing has since been heard after that press conference, with parents of the kidnapped girls still soliciting answers from the police.
Emmanuel Dovia, the Executive Secretary of the Alliance for Good Governance said the CID boss "is really contradicting herself on the airwaves and if care is not taken, she might be one of the biggest threats to our security in this nation."
In a statement, he said "said the CID boss "intended to play with our intelligence.
"We, Alliance for Good Governance, are absolutely perplexed of your useless comment you made recently."
The Democratic Forum and the Fali Foundation aloso issued separate statements expressing similar sentiments on the CID boss.
"The President must break his silence on the matter and act accordingly," said the Democratic Forum.
The group added: "We call on His Excellency Nana Akufo-Addo to, at this moment of rising tension, (in the refusal of the CID Boss to resign within 48 hours) FIRE THE CID BOSS, address the nation and assure citizens of Ghana of our safety and the commitment of his government in safeguarding same."
On the part of Fali Foundation, it said: "We are not only disappointed in the Ghana Police Service’s Criminal Investigative Department Director-General COP Maame Tiwaa Addo-Danquah, we are seriously outraged and baffled with the level of insensitivity and lack of tact in her recent comments that her earlier assurance that they knew the whereabouts of the three kidnapped Takoradi girls was actually false and she was rather misconstrued because she just wanted to give hope to the grieving families of these ladies.
"As a nongovernmental organisation that puts much premium on the girl child and women empowerment, we were delighted about her appointment to the high office of Director-General of the Criminal Investigative Department, but we must say without any equivocation that she has been very disappointing, insensitive and incompetent, and as such not worthy of emulation.
"One can only imagine the trauma and pain the affected families will be going through with these latest unfortunate revelations. The CID boss’ credibility has waned and can no longer inspire hope and her continuous stay in office doesn’t inspire confidence to the family and Ghanaians at large about her ability to get these kidnapped girls back to their families. We empathise with the affected families and we share in their pain, sorrow and frustration. We are deeply worried about the clueless and lacklustre behaviour in bringing the perpetrators to book by the police.
"Madam Tiwaa Addo-Danquah must render an unqualified apology to the affected families and Ghanaians for her gross incompetence and lack of awareness in the manner in which she gave false hope and assurances to getting these kidnapped girls rescued and reunited with their families. We call on the President to dismiss her as a matter of urgency in the spate of this glaring unprofessionalism and deliberate insensitivity. We need to restore credibility to the office of the CID and bring in competent and experienced individuals to man the office and help get our dear sisters back."