Alias

X-Factor
Founders

Angel II

Beast

Cyclops

Iceman

Marvel Girl I

X-Factor I

Angel II

Beast

Caliban

Cyclops

Iceman

Marvel Girl I

X-Factor II

Forge

Havok

Multiple Man

Polaris

Quicksilver

Random

Strong Guy

Wolfsbane

X-Factor III

Forge

Mystique

Polaris

Sabretooth

Shard

Wild Child

X-Factor IV

Butterfly

M

Multiple Man

Rictor

Strong Guy

Siryn

Wolfsbane

Occupation

X-Factor I

Privately funded mutant location and training organization.

X-Factor II

Government funded mutant super-hero team.

X-Factor III

Government funded mutant super-hero team.

X-Factor IV

Private investigation firm.

Legal Status

X-Factor I

Sanctioned by the United States government.

X-Factor II

Sanctioned and funded by the United States government.

X-Factor III

Sanctioned and funded by the United States government.

X-Factor IV

Licensed private investigation firm.

Nation or Planet of Origin

X-Factor I

New York, New York

X-Factor II

New York, New York

X-Factor III

Dallas, Texas

X-Factor IV

New York, New York

Group Alliances

X-Factor I

X-Men

X-Factor II

X-Men

X-Factor III

X-Men

X-Factor IV

X-Men

Base of Operations

X-Factor I

New York, New York

X-Factor II

New York, New York

X-Factor III

Dallas, Texas

X-Factor IV

New York, New York

Accessories

X-Factor I

Resources of Worthington Enterprises.

X-Factor II

Resources of the United States government.

X-Factor III

Resources of the United States government.

X-Factor IV

Resources common to a private investigation firm.

Common Enemies

X-Factor I

Alliance of Evil

Apocalypse

Freedom Force

Goblyn Queen

Marauders

Nanny

Orphan

(The) Right

Shadow King

X-Factor II

Random

X-Factor III

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X-Factor IV

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Regularly Appearing

X-Factor I

X-Factor Vol. 1

X-Factor II

X-Factor Vol. 1

X-Factor III

X-Factor Vol. 1

X-Factor IV

X-Factor Vol. 3

First Appearance

X-Factor I

X-Factor Vol. 1 #1 (Feb. 1986)

X-Factor II

X-Factor Vol. 1 #71 (Oct. 1991)

X-Factor III

X-Factor Vol. 1 #125 (Aug. 1996)

X-Factor IV

X-Factor Vol. 3 #1 (Jan. 2006)

Creator

X-Factor I

Bob Layton & Jackson Guice

X-Factor II

Peter David & Larry Stroman

X-Factor III

John Francis Moore & Jeff Matsuda

X-Factor IV

Peter David & Ryan Sook

Origin
After the return of Jean Grey (Marvel Girl I), the original X-Men dedicated themselves to the renewal of the mentor Professor Xavier's dream.  Much as in their early days they concentrated primarily on finding and training mutants, who were often a danger to themselves and others, in the use of their abilities.

To disguise their own mutations, while simultaneously creating support in the general human community, the five heroes perpetuated a ruse by assuming the identities of non-super-powered humans operating for a corporation known as X-Factor.

After the disbanding of the original team it's mandate and members changed over time.  Eventually many of those later members re-tasked themselves as a mutant private detective agency using the brand-name - X-Factor.