Skewing objects in SVG is the same as applying a transformation matrix to them (so, think of it as 'multiplicating the numbers with a set of other numbers').
If the object you do that to is a path, Inkscape will just crunch the numbers and change the coordinates of the path (by default).
The arrows at the end of the path are part of the path's style (they are markers). Just like the stroke, their width does not change when the coordinates of the path are changed.
If your object is a group, this approach does not work, and Inkscape will apply the calculation to the whole rendering of the group, and so, distort the group and everything in it. This is done by adding a transformation matrix to the SVG code which will be applied after the path has been rendered.
You can change this behavior in the Inkscape preferences, at Edit > Preferences > Behavior > Transforms: Store transformations: preserved. This will keep the transformation matrices and will not change the path's coordinates.
Or you can do Path > Stroke to Path to get a skewable object instead of a marker, whose shape changes when you skew it.