Quick solution - move Camera ( G-key ) with SnapTool enabled - ‘snap to face’ and ‘align rotation with target’ options ON. Then rotate around Camera local Z axis ( RZZ ) to align Camera to bottom edge of icosa face and move along local X & Y axes to fit Camera view.
Longer & perfect Camera aligning.
Idea is simple - align view to extra quad face then add camera aligned to view.
First - delete default Camera object.
Add icosa as you described , find face parallel to one of X/Y axis ( Y for me ), select it , duplicate it ( Shift-D ) , hide everything else ( Shift-H ) :
Snap 3d Cursor to face ( Shift-S>CursorToSelection ) , delete side edges of triangle but let upper vertex to stay for snap to.Enable SnapTool ( to vertex ), Extrude bottom edge ( EZ ) with Ctrl to snap to free vertex. Move new edge ( GX ) with Ctrl to snap to free vertex.Delete free vertex and select new quad face :
Hit ’ * ’ on Numpad or use Shift-V(top) menu to aling view to face then Numpad 5 to go Ortho view.Make sure you have AlignedToView option ON ( UserPreferences>EdtiMethods ). Add Camera which is aligned to face.Now you can delete additional quad face and unhide all( Alt-H ):
Select Local orientation ( Alt-Space or toolbar menu ). Rotate Camera around local Z axis ( RZZ ) 90 degree and move it a bit backward along local Z axis ( GZZ ).
As was mentioned use orthographic camera.