DIG S K Mathur, meanwhile, attempted to backtrack from his comments, saying he was totally against honour killing and that his name was being dragged into the row.
Mathur claimed that the video footage which showed him making the comments was shown "in parts" and that it was not complete.
Calling the matter serious, Akhilesh said the conduct of Mathur was unbecoming of officers holding high posts if the allegation against him is found true.
"It is a serious matter. An inquiry is being conducted into the entire episode and if found guilty action will be taken against him," Akhilesh said at a function at a city montessori school.
Later, Akhilesh said if senior officers holding higher posts do not understand the importance of their posts then action would be taken against them.
"This is true that through your medium statement of a senior officer has come...If this is the same statement, which is being aired on the channels, then it is a very wrong statement," Akhilesh told reporters.
The controversy broke out after Mathur allegedly told a man who had come to lodge a missing complaint about his sister that had he been in his place, he would have shot her for eloping.
"If officers holding high posts do not understand the seriousness of their position then action will be taken," Akhilesh said.
At the same time, he said any action taken by the government should also be projected.
"You should also show the action taken by the government after that whatever comment you have to make about law and order is acceptable," he said.
Meanwhile, Additional Director General of Police (Law and Order) Jagmohan Yadav said a report has been sought from IG Sharanpur.
Home Secretary Deepak Kumar said the government has taken a serious cognisance of the incident.
Yadav had yesterday said Mathur has denied having made the remarks.
