The SFA also said yesterday it was extending a ban on the firm taking in any new business until the end of April. The ban was first imposed last October and was due to end yesterday.
The regulator will not decide the size of its financial penalty against Fidelity Brokerage Services (FBS) until all the difficulties have been resolved.
The highly unusual action by the SFA followed complaints from clients after FBS introduced a new computer system last April. The computer problems resulted in delays with share dividend payments and the quarterly income payment into self-select personal equity plans. There were also delays in mailing clients' statements, some of which contained incorrect information.
Some complaints remain unresolved although FBS has started to pay compensation to some of its 30,000 clients. Phil West, marketing manager at FBS, declined to give precise details of the amount paid but said it was "significantly smaller" than the pounds 1m mentioned in some reports.
Compensation which had been paid to date was generally of the order of tens of pounds to each client rather than thousands, Mr West said.
The SFA said: "Good progress has been made, but further work is still required before the SFA can be confident that Fidelity Brokerage Services is offering a level of customer service which is consistently satisfactory."
During the period of the extended ban the SFA said it would monitor the customer service performance of FBS and oversee the resolution of outstanding customer complaints and enquiries.
"SFA has, in addition, placed FBS on notice that it intends to initiate disciplinary proceedings against the firm when the outstanding issues are finally resolved to the satisfaction of the authority," the SFA said.
In setting the penalty, the SFA said it would take into account the speed with which FBS dealt with the outstanding matters. The SFA said it was satisfied that FBS had adequate resources and backing to ensure investors' money and securities were not in jeopardy.
"We've made very good progress but we do recognise that there's work to be done," Mr West said.
He said FBS had significantly reduced the number of complaints - 850 clients complained originally - and upgraded the computer software.