Question

What is the transport of material out of a cell by means of a sac or vesicle that first engulfs the material and then is extruded through an opening in the cell membrane?

Answer 1

Exocytosis occurred, where a vesicle or liposome is transported through the cell's cytoskeleton to the cell surface membrane to fuse with it, releasing the contents from within the vesicle and into the extracellular environment.
It does not matter if the vesicle or liposome had previously fused with a lysosome previously.
So long as it is a vesicle/liposome, which has a phospholipid bilayer membrane (I.e. the same as the cell surface membrane), and fusion occurs, it is exocytosis.

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