Inverse laplace transform of step function

Questions
1

I tried to do

InverseLaplaceTransform[HeavisideTheta[s],s,t]

and didn’t get anything.

any suggestions?

2

Observe

In[15]:= InverseLaplaceTransform[LaplaceTransform[HeavisideTheta[s], s, t], t, s]
Out[15]= 1

despite the statement in the reference: LaplaceTransform and InverseLaplaceTransform are mutual inverse

In[16]:= InverseLaplaceTransform[LaplaceTransform[y[s], s, t], t, s]
Out[16]= y[s]

Does this entitle anybody to consider

In[4]:= InverseLaplaceTransform[1, s, t]
Out[4]= DiracDelta[t]

DiracDelta as InverseLaplaceTransform of HeavisideTheta?

Rather not, DiracDelta is the InverseLaplaceTransform of 1, but not of HeavisideTheta. If Mathematica tells the truth, an InverseLaplaceTransform of HeavisideTheta does not exist, at least I did not find one in the books.

What do you thing the result of

In[1]:= InverseLaplaceTransform[HeavisideTheta[s], s, t]
Out[1]= InverseLaplaceTransform[HeavisideTheta[s], s, t]

should be?