fortführen
to continue
To carry on or continue an activity, conversation, or process that has already begun.
Wir werden die Diskussion in der nächsten Sitzung fortführen.
We will continue the discussion in the next meeting.

to continue / lead away
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to continue
To carry on or continue an activity, conversation, or process that has already begun.
Wir werden die Diskussion in der nächsten Sitzung fortführen.
We will continue the discussion in the next meeting.

'Fortführen' (to continue / lead away) is a separable verb. The prefix 'fort-' comes at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Der neue Chef führt die Projekte seines Vorgängers fort.
The new boss continues his predecessor's projects.
The verb consists of the prefix 'fort-' (away, onward) and 'führen' (to lead). Depending on the context, it therefore means to 'lead something onward' (continue) or to 'lead someone away' (take away).
Lassen Sie uns das Gespräch nach der Pause fortführen.
Let us continue the conversation after the break.
In the meaning 'fortsetzen' (to continue), the verb is often used with abstract nouns. Typical combinations are 'eine Arbeit fortführen' (to continue a piece of work), 'eine Tradition fortführen' (to continue a tradition), or 'ein Projekt fortführen' (to continue a project).
Nach dem Tod des Gründers führten seine Kinder die Firma fort.
After the founder's death, his children continued the company.
You can recognize this meaning by the fact that a person or a movable object is the object. The context is often formal and describes an action by figures of authority such as the police.
Die Wachen führten den Gefangenen aus seiner Zelle fort.
The guards led the prisoner away from his cell.
to end
'beenden' and 'fortführen' are opposites when something has already begun. 'beenden' means that you do not continue and that you bring something to an end. 'fortführen' means that you continue with the same activity, conversation, or process. 'fortführen' sounds somewhat more formal than 'weitermachen' (to continue / carry on), but it fits well with projects, conversations, and work. The simple difference is: beenden = to stop or finish, fortführen = to continue.

Mein Freund führt das Projekt im Zug weiter fort.
My friend continues the project on the train.
In der Schule haben wir das Gespräch über die Klassenfahrt beendet.
At school, we ended the conversation about the class trip.