I understand もし寝過ごせば、起こしてください is wrong. I believe instead it should be
寝過ごしたら、起こしてください。

In "A Dictionary of Basic Japanese Grammar" it says the second part of the
sentence "...can be a command, a request or a suggestion... in this case,
however, [the first part] can not be an action".

As the second part of the above sentence is a request then I take it
寝過ごす「ねすごす」is an action? If I'm right can anyone give more examples of action
verbs as I would not have thought of oversleeping as an action like coming and
going. Is their any connection between transitivity and intransitivity of
verbs and whether they are classed as an action?

Thanks
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verbs
Action verbs and conditional ～ば
I think what the dictionary meant by "not an action" means that the action is
not realised yet. For example anything in `~ば` means "in the case that ~"
which is hypothetical.

S1 ば S2 makes a condition that S2 will hold true under S1.

This makes for a generic conditional statement that cannot bear a unverifiable
volition. How I derived this conclusion is because sentences such as
`安ければ買います。`((I) will buy if it's cheap) is possible. It is also possible if
the person is someone you can empathise with(i.e. share the viewpoint
accurately enough). But once it becomes a request it cannot become a true
condition of "If A is true, B is true".

And for `~た(な)ら(ば)`, observe that it is actually the "past tense" for `~ば`
which conveys a subjunctive past. It presupposes that the action(S1) is
realised, then S2. This makes for a statement that once S1 completes, do S2.

This can be used for making requests. "If A has happened, please do B".
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I would say that anything that happens (is realized) at some point is an
"action" in this context. The action 寝過ごす is realized when the train you're on
reaches your station, but you're sleeping. Some other verbs like this would be
起きる, 寝る, 食べる etc. So you _cannot_ say

  * 彼が起きれば、私に電話をするように伝えて下さい <\- WRONG!!

"Non actions" in this context would be e.g. adjectives, verbs that specify
some state, like いる or ～て／でいる forms of other verbs, like 起きている, 寝ている. So you
_can_ say

  * 荷物が重ければ、私に電話をして下さい Please call me if the baggage is heavy
  * 彼が家にいれば、私に電話をするように伝えて下さい If he is at home, please ask him to call me
  * 彼が起きていれば、私に電話をするように伝えて下さい If he is awake, please ask him to call me

To try to answer your second question, I don't think transitivity has much to
do with it. As I've already argued with the 起きる vs 起きている example, there are
cases where it's more how the verb is used than the type of the verb itself.
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