If it wants to do business here in CA, it needs a qualify as a "foreign" corporation and pay Franchise Board Tax and designate a CA agent for service of process, and then then register for a CA dba.
See a business lawyer to help you get your business properly organized.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
And, if you use the same business name as that associated with your foreign corporate registration in California, then you will not need to register as a fictious business name.
My response does not constitute legal advice, as I do not know all of the relevant facts of your case, and I do not legally represent you. Although I strive to make sure the information I provide is generally accurate and useful, you should promptly consult an able lawyer who can learn the unique details of your case more completely in a confidential relationship to ensure that the information I provide, and your interpretation of it, is appropriate to your particular situation.
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That is secondary. First, the foreign corporation will need to register with the California Secretary of State as a foreign entity doing business in California (and thereby subjecting it to California taxation and regulation; in some instances, in may be better just to form a new California business entity). Depending on the circumstances, BOE, EDD, business license, and other registrations may be needed.
In California, fictitious business names are filed in at least one of the counties in which the business operates. Filing such a DBA alone does not entitle an out-of-state business entity to do business in California, under any name. Here's some more information about California fictitious business names, from my website: http://www.incorporatecalifornia.com/dba-fbn.html
It would appear from your question that you would be well advised to seek the assistance of an experienced California business attorney to help ensure the success of your new business venture.
Dislaimer: This does not constitute legal advice or form an attorney-client relationship.