Funeral directors in England, Wales and Northern Ireland are unregulated, with anyone being able to set up in the trade, but the rules in Scotland are different.
In its first review of the rules, the CMA said that price rises had stalled, with firms charging an average of just over 2,600 for funeral services.
It said that 100 funeral directors - 5% of the sector -- appeared not to be keeping to the rules on price transparency and it would be writing enforcement letters to those businesses.
If they failed to improve, they would be named and shamed by the CMA, and faced legal action to make sure they complied with the rules.
The CMA says that funeral providers should give their customers all the information they need to know when they lose a loved one.
We are ramping up enforcement actions to make sure that the minority of funeral providers who are not open with bereaved families face the consequences. He said that the CMA should revisit the idea of price controls, something the authority temporarily ruled out due to the impact of the Covid epidemic on the sector.
Funeral prices, and profit margins, were still too high and more should be done to bring them down, he said.
Pre-paid funeral plans allow people to plan and pay for their own funeral in advance. Since last year, these have been regulated by the Financial Conduct Authority.
There is a state safety net, the Funeral Expenses Payment, available to people in England and Wales on certain benefits, but critics say it leaves a shortfall that has to be picked up by families. It also operates in Northern Ireland but is claimed differently.