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I have this mesh:

enter image description here

I want it to become like:

enter image description here

But if I simply extrude inward, it becomes:

enter image description here

I know in this simplified case I could just extrude the square in the center and scale them downwards. But I'd like to know if there a "correct" way to do the opposite of extrusion.

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Extrude Manifold

In your last image, the infinitely thin areas created around the edge are called non-manifold. Blender now has a tool called Extrude Manifold which will automatically fix this issue.

To use this tool, press ⎇ AltE for the extrude menu and select it.

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Upon dragging your cursor down, you will see the satisfying result:

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    $\begingroup$ day successful. Learnt again something from you! ;) +1 $\endgroup$
    – Chris
    Jun 22, 2021 at 4:06
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    $\begingroup$ I almost forgot about this tool when I was about to start answering. Watching Blender Today live streams paid off. 😉 $\endgroup$
    – HISEROD
    Jun 22, 2021 at 4:11
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    $\begingroup$ I didn't expect there is a tool for this exact reason. Many thanks! $\endgroup$
    – Lai Yu-Hsuan
    Jun 22, 2021 at 4:13
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    $\begingroup$ ... Implemented because everyone was getting SketchUp-envy :) $\endgroup$
    – Robin Betts ♦
    Jun 22, 2021 at 6:04
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    $\begingroup$ @LaiYu-Hsuan The only thing you have to be aware of: you have this edge loop below the top. If you Extrude Manifold past this loop, it will look the same like your normal extrude, the border faces will stay like a "wall" around the inner extrude. Or, if you do that but then move the faces back above that edge loop with G > Z, and then try to Extrude Manifold again, it will cause some weird behaviour. $\endgroup$
    – Gordon Brinkmann
    Jun 22, 2021 at 7:55
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Select these faces, then press G -> Z and move down.

Then select middle face and extrude up.

Select the middle face.

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